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# Circles Lecture - Part 1

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## Circles Lecture - Part 1Presentation Transcript

• Circles Mathematics 4 August 10, 2011Mathematics 4 () Circles August 10, 2011 1 / 17
• Review of Completing the SquareCompleting the SquareExpress the following quadratic expressions as (y − k) = a(x − h)2 . Mathematics 4 () Circles August 10, 2011 2 / 17
• Review of Completing the SquareCompleting the SquareExpress the following quadratic expressions as (y − k) = a(x − h)2 . 1 x2 − y − 12x + 7 = 0 Mathematics 4 () Circles August 10, 2011 2 / 17
• Review of Completing the SquareCompleting the SquareExpress the following quadratic expressions as (y − k) = a(x − h)2 . 1 x2 − y − 12x + 7 = 0 → (y + 29) = (x − 6)2 Mathematics 4 () Circles August 10, 2011 2 / 17
• Review of Completing the SquareCompleting the SquareExpress the following quadratic expressions as (y − k) = a(x − h)2 . 1 x2 − y − 12x + 7 = 0 → (y + 29) = (x − 6)2 2 2x2 − 5x − y − 3 = 0 Mathematics 4 () Circles August 10, 2011 2 / 17
• Review of Completing the SquareCompleting the SquareExpress the following quadratic expressions as (y − k) = a(x − h)2 . 1 x2 − y − 12x + 7 = 0 → (y + 29) = (x − 6)2 2 2x2 − 5x − y − 3 = 0 49 →y+ 8 = 2(x − 5 )2 4 Mathematics 4 () Circles August 10, 2011 2 / 17
• Review of Completing the SquareCompleting the SquareExpress the following expressions as (x − h)2 + (y − k)2 = r, where h, k,and r are constants. Mathematics 4 () Circles August 10, 2011 3 / 17
• Review of Completing the SquareCompleting the SquareExpress the following expressions as (x − h)2 + (y − k)2 = r, where h, k,and r are constants. 1 x2 + y 2 + 2x − 8y + 4 = 0 Mathematics 4 () Circles August 10, 2011 3 / 17
• Review of Completing the SquareCompleting the SquareExpress the following expressions as (x − h)2 + (y − k)2 = r, where h, k,and r are constants. 1 x2 + y 2 + 2x − 8y + 4 = 0 → (x + 1)2 + (y − 4)2 = 13 Mathematics 4 () Circles August 10, 2011 3 / 17
• Review of Completing the SquareCompleting the SquareExpress the following expressions as (x − h)2 + (y − k)2 = r, where h, k,and r are constants. 1 x2 + y 2 + 2x − 8y + 4 = 0 → (x + 1)2 + (y − 4)2 = 13 2 9x2 + 9y 2 + 6x − 12y + 5 = 63 Mathematics 4 () Circles August 10, 2011 3 / 17
• Review of Completing the SquareCompleting the SquareExpress the following expressions as (x − h)2 + (y − k)2 = r, where h, k,and r are constants. 1 x2 + y 2 + 2x − 8y + 4 = 0 → (x + 1)2 + (y − 4)2 = 13 2 9x2 + 9y 2 + 6x − 12y + 5 = 63 → (x + 3 )2 + (y − 2 )2 = 7 1 3 Mathematics 4 () Circles August 10, 2011 3 / 17
• Circles What is a circle? Mathematics 4 () Circles August 10, 2011 4 / 17
• CirclesDeﬁnition of CirclesA circle is a set of all points (locus) that are the same distance from agiven point. Mathematics 4 () Circles August 10, 2011 5 / 17
• CirclesDeﬁnition of CirclesA circle is a set of all points (locus) that are the same distance from agiven point.Terminology Mathematics 4 () Circles August 10, 2011 5 / 17
• CirclesDeﬁnition of CirclesA circle is a set of all points (locus) that are the same distance from agiven point.Terminology same distance → radius Mathematics 4 () Circles August 10, 2011 5 / 17
• CirclesDeﬁnition of CirclesA circle is a set of all points (locus) that are the same distance from agiven point.Terminology same distance → radius given point → center Mathematics 4 () Circles August 10, 2011 5 / 17
• The Standard Form of the Circle EquationThe Distance FormulaThe distance between two points (x1 , y1 ) and (x2 , y2 ) in the Cartesianplane is given by: d= (x2 − x1 )2 + (y2 − y1 )2 Mathematics 4 () Circles August 10, 2011 6 / 17
• The Standard Form of the Circle EquationThe Distance FormulaUse the distance formula to relate the radius with the center of the circle. r= (x − h)2 + (y − k)2 (1) Mathematics 4 () Circles August 10, 2011 7 / 17
• The Standard Form of the Circle EquationStandard Form/Center-Radius FormGiven a circle with center at (h, k) and having a radius r, the center radiusform of the circle equation is given by: (x − h)2 + (y − k)2 = r2 Mathematics 4 () Circles August 10, 2011 8 / 17
• Graphing ExamplesGraph x2 + y 2 = 10. Label center, radius, and any intercepts. Mathematics 4 () Circles August 10, 2011 9 / 17
• Graphing ExamplesGraph x2 + y 2 = 10. Label center, radius, and any intercepts. √ √ x-intercepts → 10, − 10 Mathematics 4 () Circles August 10, 2011 9 / 17
• Graphing ExamplesGraph x2 + y 2 = 10. Label center, radius, and any intercepts. √√ x-intercepts → 10, − 10 √ √ y-intercepts → 10, − 10 Mathematics 4 () Circles August 10, 2011 9 / 17
• Graphing ExamplesGraph (x + 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts. Mathematics 4 () Circles August 10, 2011 10 / 17
• Graphing ExamplesGraph (x + 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts. √ √ x-intercepts → −3 + 5, −3 − 5 Mathematics 4 () Circles August 10, 2011 10 / 17
• Graphing ExamplesGraph (x + 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts. √ √ x-intercepts → −3 + 5, −3 − 5 y-intercepts → 2 Mathematics 4 () Circles August 10, 2011 10 / 17
• Problems on CirclesExample 1What is the equation of a circle with radius 5, centered on the origin?Graph this circle. Mathematics 4 () Circles August 10, 2011 11 / 17
• Problems on CirclesExample 1What is the equation of a circle with radius 5, centered on the origin?Graph this circle. Mathematics 4 () Circles August 10, 2011 11 / 17
• Problems on CirclesExample 1What is the equation of a circle with radius 5, centered on the origin?Graph this circle. x2 + y 2 = 25 Mathematics 4 () Circles August 10, 2011 11 / 17
• Problems on CirclesExample 1What is the equation of a circle with radius 5, centered on the origin?Graph this circle. x2 + y 2 = 25 Mathematics 4 () Circles August 10, 2011 11 / 17
• Problems on CirclesExample 2Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts. Mathematics 4 () Circles August 10, 2011 12 / 17
• Problems on CirclesExample 2Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts. Mathematics 4 () Circles August 10, 2011 12 / 17
• Problems on CirclesExample 2Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts. (x + 3)2 + (y − 2)2 = 25 Mathematics 4 () Circles August 10, 2011 12 / 17
• Problems on CirclesExample 2Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts. (x + 3)2 + (y − 2)2 = 25 √x-intercepts : −3 ± 21, y-intercepts: 6, −2 Mathematics 4 () Circles August 10, 2011 12 / 17
• Problems on CirclesExample 3Find an equation of a circle with a diameter whose endpoints are atP1 (7, −3) and P2 (1, 7). Mathematics 4 () Circles August 10, 2011 13 / 17
• Problems on CirclesExample 3Find an equation of a circle with a diameter whose endpoints are atP1 (7, −3) and P2 (1, 7). Mathematics 4 () Circles August 10, 2011 13 / 17
• Problems on CirclesExample 3Find an equation of a circle with a diameter whose endpoints are atP1 (7, −3) and P2 (1, 7). center: Use the midpoint formula x1 + x2 y1 + y2 → (h, k) = , = (4, 2) 2 2 Mathematics 4 () Circles August 10, 2011 13 / 17
• Problems on CirclesExample 3Find an equation of a circle with a diameter whose endpoints are atP1 (7, −3) and P2 (1, 7). center: Use the midpoint formula x1 + x2 y1 + y2 → (h, k) = , = (4, 2) 2 2 radius: Distance from one endpoint to the center √ → r = (x1 − h)2 + (y1 − k)2 = 34 Mathematics 4 () Circles August 10, 2011 13 / 17
• Problems on CirclesExample 3Find an equation of a circle with a diameter whose endpoints are atP1 (7, −3) and P2 (1, 7). center: Use the midpoint formula x1 + x2 y1 + y2 → (h, k) = , = (4, 2) 2 2 radius: Distance from one endpoint to the center √ → r = (x1 − h)2 + (y1 − k)2 = 34 (x − 4)2 + (y − 2)2 = 34 Mathematics 4 () Circles August 10, 2011 13 / 17
• The General Form of the Circle EquationRewriting the answer to the previous problem: (x − 4)2 + (y − 2)2 = 34 Mathematics 4 () Circles August 10, 2011 14 / 17
• The General Form of the Circle EquationRewriting the answer to the previous problem: (x − 4)2 + (y − 2)2 = 34 → x2 + y 2 − 8x − 4y − 14 = 0This is called the General Form of the Circle Equation. Mathematics 4 () Circles August 10, 2011 14 / 17
• The General Form of the Circle EquationThe General Form of the Circle Equation Ax2 + Ay 2 + Cx + Dy + E = 0The x2 and y 2 terms should have identical coeﬃcients. Mathematics 4 () Circles August 10, 2011 15 / 17
• Problems on CirclesExample 4Find the center and radius of the circle with the equationx2 + y 2 + 4x − 6y + 5 = 0. Mathematics 4 () Circles August 10, 2011 16 / 17
• Problems on CirclesExample 4Find the center and radius of the circle with the equationx2 + y 2 + 4x − 6y + 5 = 0. (x2 + 4x) + (y 2 − 6y) = −5 Mathematics 4 () Circles August 10, 2011 16 / 17
• Problems on CirclesExample 4Find the center and radius of the circle with the equationx2 + y 2 + 4x − 6y + 5 = 0. (x2 + 4x) + (y 2 − 6y) = −5 (x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9 Mathematics 4 () Circles August 10, 2011 16 / 17
• Problems on CirclesExample 4Find the center and radius of the circle with the equationx2 + y 2 + 4x − 6y + 5 = 0. (x2 + 4x) + (y 2 − 6y) = −5 (x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9 (x + 2)2 + (y − 3)2 = 8 Mathematics 4 () Circles August 10, 2011 16 / 17
• Problems on CirclesExample 4Find the center and radius of the circle with the equationx2 + y 2 + 4x − 6y + 5 = 0. (x2 + 4x) + (y 2 − 6y) = −5 (x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9 (x + 2)2 + (y − 3)2 = 8 Mathematics 4 () Circles August 10, 2011 16 / 17
• Problems on CirclesExample 4Find the center and radius of the circle with the equationx2 + y 2 + 4x − 6y + 5 = 0. (x2 + 4x) + (y 2 − 6y) = −5 (x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9 (x + 2)2 + (y − 3)2 = 8 C(−2, 3)√ √ r= 8=2 2 Mathematics 4 () Circles August 10, 2011 16 / 17
• Problems on CirclesMore examples Mathematics 4 () Circles August 10, 2011 17 / 17
• Problems on CirclesMore examples 1 Find the standard equation of a circle with center at (1, 5) and passes through (7, 2). Mathematics 4 () Circles August 10, 2011 17 / 17
• Problems on CirclesMore examples 1 Find the standard equation of a circle with center at (1, 5) and passes through (7, 2). 2 Find the area of the circle with equation x2 + y 2 + 8x − 12y − 14. Mathematics 4 () Circles August 10, 2011 17 / 17
• Problems on CirclesMore examples 1 Find the standard equation of a circle with center at (1, 5) and passes through (7, 2). 2 Find the area of the circle with equation x2 + y 2 + 8x − 12y − 14. 3 Find the general equation of the circle tangent to both axes, whose center is in QII, and whose radius is 4. Mathematics 4 () Circles August 10, 2011 17 / 17