32 conic sections, circles and completing the square
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Conic Sections
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Conic SectionsOne way to study a solid is to slice it open.
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown. A right circular cone
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown. A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown.A Horizontal Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown.A Horizontal Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown.A ModeratelyTilted Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown.A ModeratelyTilted Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown.A Horizontal SectionA ModeratelyTilted SectionCircles andellipsis areenclosed. A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown. A Parallel–Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown. A Parallel–Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown. An Cut-away Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown. An Cut-away Section A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsOne way to study a solid is to slice it open. The exposed areaof the sliced solid is called a cross sectional area.Conic sections are the borders of the cross sectional areas ofa right circular cone as shown.A Horizontal SectionA Moderately A Parallel–SectionTilted Section An Cut-awayCircles and Sectionellipsis areenclosed. Parabolas and hyperbolas are open. A right circular cone and conic sections (wikipedia “Conic Sections”)
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Conic SectionsWe summarize the four types of conics sections here.Circles EllipsesParabolas Hyperbolas
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Conic SectionsWe summarize the four types of conics sections here.Circles EllipsesParabolas HyperbolasBesides their differences in visual appearance and the mannersthey reside inside the cone, there are many reasons, that havenothing to do with cones, that the conic sections are groupedinto four groups.
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Conic SectionsWe summarize the four types of conics sections here.Circles EllipsesParabolas HyperbolasBesides their differences in visual appearance and the mannersthey reside inside the cone, there are many reasons, that havenothing to do with cones, that the conic sections are groupedinto four groups. One way is to use distance relations to classifythem.
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Conic SectionsWe summarize the four types of conics sections here.Circles EllipsesParabolas HyperbolasBesides their differences in visual appearance and the mannersthey reside inside the cone, there are many reasons, that havenothing to do with cones, that the conic sections are groupedinto four groups. One way is to use distance relations to classifythem. We use the circles and the ellipsis as examples.
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r. C
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r. C
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r. C
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r. C Hence a dog tied to a post would mark off a circular track.
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), F1 F2
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant. F1 F2
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant. QFor example, if P, Q, and R Pare points on a ellipse, F1 F2 R
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant. QFor example, if P, Q, and R Pare points on a ellipse, then p2 p1p1 + p2 F1 F2 R
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant. QFor example, if P, Q, and R P p2 q1are points on a ellipse, then q2 p1p1 + p2 F1 F2= q1 + q2 R
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant. QFor example, if P, Q, and R P p2 q1are points on a ellipse, then q2 p1p1 + p2 F1 F2= q1 + q2= r1 + r 2 r1 r2= a constant R
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant. QFor example, if P, Q, and R P p2 q1are points on a ellipse, then q2 p1p1 + p2 F1 F2= q1 + q2= r1 + r 2 r1 r2= a constant Hence a dog leashed by a ring R to two posts would mark off an elliptical track.
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CirclesGiven a fixed point C, a circle is theset of points whose distances to C is r ra fixed constant r.The equal-distance r is called the Cradius and the point C is called thecenter of the circle.Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant. QFor example, if P, Q, and R P p2 q1are points on a ellipse, then q2 p1p1 + p2 F1 F2= q1 + q2= r1 + r 2 r1 r2= a constantLikewise parabolas and hyperbolas Rmay be defined using relations of distance measurements.
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Conic SectionsThe second reason that we group the conic sections into fourtypes is algebraic, i.e. the equations related to graphs of theconic sections can easily be sorted into the above four types
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Conic SectionsThe second reason that we group the conic sections into fourtypes is algebraic, i.e. the equations related to graphs of theconic sections can easily be sorted into the above four typesRecall that straight linesare the graphs of1st degree equationsAx + By = Cwhere A, B, C, are numbers.
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Conic SectionsThe second reason that we group the conic sections into fourtypes is algebraic, i.e. the equations related to graphs of theconic sections can easily be sorted into the above four typesRecall that straight linesare the graphs of1st degree equationsAx + By = C y = –1 x=1 y+x=1 Linear graphswhere A, B, C, are numbers.
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Conic SectionsThe second reason that we group the conic sections into fourtypes is algebraic, i.e. the equations related to graphs of theconic sections can easily be sorted into the above four typesRecall that straight linesare the graphs of1st degree equationsAx + By = C y = –1 x=1 y+x=1 Linear graphswhere A, B, C, are numbers.Conic sections are the graphs of 2nd degree equations inx and y.
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Conic SectionsThe second reason that we group the conic sections into fourtypes is algebraic, i.e. the equations related to graphs of theconic sections can easily be sorted into the above four typesRecall that straight linesare the graphs of1st degree equationsAx + By = C y = –1 x=1 y+x=1 Linear graphswhere A, B, C, are numbers.Conic sections are the graphs of 2nd degree equations inx and y. In particular, the conic sections that are parallel to theaxes (not tilted) have equations of the formAx2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.
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Conic SectionsThe second reason that we group the conic sections into fourtypes is algebraic, i.e. the equations related to graphs of theconic sections can easily be sorted into the above four typesRecall that straight linesare the graphs of1st degree equationsAx + By = C y = –1 x=1 y+x=1 Linear graphswhere A, B, C, are numbers.Conic sections are the graphs of 2nd degree equations inx and y. In particular, the conic sections that are parallel to theaxes (not tilted) have equations of the formAx2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.The algebraic technique that enables us to sort these 2nddegree equations into four groups of conic sections is called"completing the square".
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Conic SectionsThe second reason that we group the conic sections into fourtypes is algebraic, i.e. the equations related to graphs of theconic sections can easily be sorted into the above four typesRecall that straight linesare the graphs of1st degree equationsAx + By = C y = –1 x=1 y+x=1 Linear graphswhere A, B, C, are numbers.Conic sections are the graphs of 2nd degree equations inx and y. In particular, the conic sections that are parallel to theaxes (not tilted) have equations of the formAx2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.The algebraic technique that enables us to sort these 2nddegree equations into four groups of conic sections is called"completing the square". We will apply this method to thecircles but only summarize the results about the other ones.
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center. r r center
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center.The radius and the center completely determine the circle. r r center
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center.The radius and the center completely determine the circle.Let (h, k) be the center of acircle and r be the radius. (h, k) r
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center.The radius and the center completely determine the circle.Let (h, k) be the center of acircle and r be the radius.Suppose (x, y) is a point on (x, y)the circle, then the distancebetween (x, y) and the center (h, k) ris r.
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center.The radius and the center completely determine the circle.Let (h, k) be the center of acircle and r be the radius.Suppose (x, y) is a point on (x, y)the circle, then the distancebetween (x, y) and the center (h, k) ris r. Hence,r = √ (x – h)2 + (y – k)2
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center.The radius and the center completely determine the circle.Let (h, k) be the center of acircle and r be the radius.Suppose (x, y) is a point on (x, y)the circle, then the distancebetween (x, y) and the center (h, k) ris r. Hence,r = √ (x – h)2 + (y – k)2 orr2 = (x – h)2 + (y – k)2
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center.The radius and the center completely determine the circle.Let (h, k) be the center of acircle and r be the radius.Suppose (x, y) is a point on (x, y)the circle, then the distancebetween (x, y) and the center (h, k) ris r. Hence,r = √ (x – h)2 + (y – k)2 orr2 = (x – h)2 + (y – k)2This is called the standard form of circles.
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CirclesA circle is the set of all the points that have equal distance r,called the radius, to a fixed point C which is called the center.The radius and the center completely determine the circle.Let (h, k) be the center of acircle and r be the radius.Suppose (x, y) is a point on (x, y)the circle, then the distancebetween (x, y) and the center (h, k) ris r. Hence,r = √ (x – h)2 + (y – k)2 orr2 = (x – h)2 + (y – k)2This is called the standard form of circles. Given an equationof this form, we can easily identify the center and the radius.
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Circlesr is the radius must be “ – ” r2 = (x – h)2 + (y – k)2
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Circlesr is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the center
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Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the centerExample A. Write the equation (–1, 8)of the circle as shown. (–1, 3)
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Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the centerExample A. Write the equation (–1, 8)of the circle as shown.The center is (–1, 3) and theradius is 5. (–1, 3)
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Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the centerExample A. Write the equation (–1, 8)of the circle as shown.The center is (–1, 3) and theradius is 5. (–1, 3)Hence the equation is:52 = (x – (–1))2 + (y – 3)2
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Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the centerExample A. Write the equation (–1, 8)of the circle as shown.The center is (–1, 3) and theradius is 5. (–1, 3)Hence the equation is:52 = (x – (–1))2 + (y – 3)2 or25 = (x + 1)2 + (y – 3 )2
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Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the centerExample A. Write the equation (–1, 8)of the circle as shown.The center is (–1, 3) and theradius is 5. (–1, 3)Hence the equation is:52 = (x – (–1))2 + (y – 3)2 or25 = (x + 1)2 + (y – 3 )2In particular a circle centered atthe origin has an equation ofthe form x2 + y2 = r2
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CirclesExample B. Identify the center andthe radius of 16 = (x – 3)2 + (y + 2)2.Label the top, bottom, left and rightmost points. Graph it.
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CirclesExample B. Identify the center andthe radius of 16 = (x – 3)2 + (y + 2)2.Label the top, bottom, left and rightmost points. Graph it.Put 16 = (x – 3)2 + (y + 2)2 into thestandard form: 42 = (x – 3)2 + (y – (–2))2
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CirclesExample B. Identify the center andthe radius of 16 = (x – 3)2 + (y + 2)2.Label the top, bottom, left and rightmost points. Graph it.Put 16 = (x – 3)2 + (y + 2)2 into thestandard form: 42 = (x – 3)2 + (y – (–2))2Hence r = 4, center = (3, –2)
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CirclesExample B. Identify the center andthe radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)Label the top, bottom, left and rightmost points. Graph it.Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6)Hence r = 4, center = (3, –2)
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CirclesExample B. Identify the center andthe radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)Label the top, bottom, left and rightmost points. Graph it.Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6)Hence r = 4, center = (3, –2)When equations are not in the standard form, we have torearrange them into the standard form. We do this by"completing the square".
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CirclesExample B. Identify the center andthe radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)Label the top, bottom, left and rightmost points. Graph it.Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6)Hence r = 4, center = (3, –2)When equations are not in the standard form, we have torearrange them into the standard form. We do this by"completing the square". To complete the square means toadd a number to an expression so the sum is a perfectsquare.
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CirclesExample B. Identify the center andthe radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)Label the top, bottom, left and rightmost points. Graph it.Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6)Hence r = 4, center = (3, –2)When equations are not in the standard form, we have torearrange them into the standard form. We do this by"completing the square". To complete the square means toadd a number to an expression so the sum is a perfectsquare. This procedure is the main technique in dealing with2nd degree equations.
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square,
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2b. y2 + 12y + (12/2)2
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2b. y2 + 12y + (12/2)2
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2The following are the steps in putting a 2nd degree equationinto the standard form.
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2The following are the steps in putting a 2nd degree equationinto the standard form.1. Group the x2 and the x-terms together, group the y2 and y terms together, and move the number term to the other side of the equation.
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CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expressionmakes the expression a perfect square, i.e. x2 + bx + (b/2)2is the perfect square (x + b/2)2.Example C. Fill in the blank to make a perfect square.a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2The following are the steps in putting a 2nd degree equationinto the standard form.1. Group the x2 and the x-terms together, group the y2 and y terms together, and move the number term to the other side of the equation.2. Complete the square for the x-terms and for the y-terms. Make sure to add the necessary numbers to both sides.
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32Hence the center is (3, –6),and radius is 3.
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CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32Hence the center is (3, –6),and radius is 3. (3, –6),
86.
CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32Hence the center is (3, –6), (3, –3),and radius is 3. (0, –6), (6, –6), (3, –6), (–9, –6)
87.
CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32Hence the center is (3, –6), (3, –3),and radius is 3.The Completing-the-Squaremethod is the basic method for (0, –6), (6, –6), (3, –6),handling 2nd degree problems. (–9, –6)
88.
CirclesExample E. Use completing the square to find the centerand radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,left and right most points. Graph it.We use completing the square to put the equation into thestandard form:x2 – 6x + + y2 + 12y + = –36 complete the squaresx2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32Hence the center is (3, –6), (3, –3),and radius is 3.The Completing-the-Squaremethod is the basic method for (0, –6), (6, –6), (3, –6),handling 2nd degree problems.We summarize the hyperbolaand parabola below. (–9, –6)
90.
HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.
91.
HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.
92.
HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown C A B
93.
HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena 1 – a2 C A a1 a2 B
94.
HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 = b1 – b2 C A a1 a2 b2 B b1
95.
HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 = b1 – b2 = c2 – c1 = constant. C c2 A a1 c1 a2 b2 B b1
96.
ParabolasFinally, we illustrate the definition that’s based on distancemeasurements of the parabolas.Given a fixed point F, and a line L, the points that are of equaldistance from F the line L is a parabola.Hence a = A, b = B, c = C as shown below.For more information, see:http://en.wikipedia.org/wiki/Parabola F L
97.
ParabolasFinally, we illustrate the definition that’s based on distancemeasurements of the parabolas.Given a fixed point F, and a line L, the points that are of equaldistance from F the line L is a parabola.Hence a = A, b = B, c = C as shown below.For more information, see:http://en.wikipedia.org/wiki/Parabola P1 F a A L
98.
ParabolasFinally, we illustrate the definition that’s based on distancemeasurements of the parabolas.Given a fixed point F, and a line L, the points that are of equaldistance from F the line L is a parabola.Hence a = A, b = B, c = C as shown below.For more information, see:http://en.wikipedia.org/wiki/Parabola P1 F a b P2 A B L
99.
ParabolasFinally, we illustrate the definition that’s based on distancemeasurements of the parabolas.Given a fixed point F, and a line L, the points that are of equaldistance from F the line L is a parabola.Hence a = A, b = B, c = C as shown below.For more information, see:http://en.wikipedia.org/wiki/Parabola a b c A B C
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