Discuss how the radius can be drawn from the center to any point on the circle.
Stress that in a linear equation, no variables are squared. In the parabola equation only one variable is squared, while two are squared in the circle equation.
Ax+BY=C Stress that linear equations have exponents of 1 on both the x and the y. With parabolas, either the x or the y has an exponent of 2. With circles, both x and y terms are squared. Also with circles, the coefficients (or denominators) are the same for x 2 and y 2 (You will see in another lesson that that is not true for ellipses.)
Transcript
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What are conics? <ul><li>Conics, or conic sections, are the intersection of a plane with an infinite double cone. If that plane cuts both cones, it is a hyperbola. If it is parallel to the edge of the cone, you get a parabola. If neither is the case, it is an ellipse. The ellipse is also a circle if the plane is perpendicular to the altitude of the cone. </li></ul>
<ul><li>Review: The geometric definition relies on a cone and a plane intersecting it </li></ul><ul><li>Algebraic definition: a set of points in the plane that are equidistant from a fixed point on the plane (the center). </li></ul>
<ul><li>Find the distance from the center of the circle (h,k) to any point on the circle (represented by (x,y)). This is the radius of the circle. </li></ul><ul><li>Review the distance formula: </li></ul><ul><li>Substitute in the values. </li></ul><ul><li>Square both sides to get </li></ul><ul><li>the general form of a </li></ul><ul><li>circle in center-radius form. </li></ul>y x r (h,k) (x,y)
<ul><li>Both variables are squared. </li></ul><ul><li>Equation of a circle in center-radius form: </li></ul><ul><li>What makes the circle different from the a line? </li></ul><ul><li>What makes the circle different from the parabola? </li></ul>
<ul><li>4. Write the equation of a circle centered at (2,-7) and having a radius of 5. </li></ul><ul><li>(x - 2) 2 + (y + 7) 2 = 25 </li></ul><ul><li>5. Describe (x - 2) 2 + (y + 1) 2 = 0 </li></ul><ul><li>A point at (2,-1) </li></ul><ul><li>6. Describe (x + 1) 2 + (y - 3) 2 = -1 </li></ul><ul><li>No graph </li></ul>
<ul><li>7. Write the equation of a circle whose diameter is the line segment joining A(-3,-4) and B(4,3). </li></ul><ul><li>What must you find first? </li></ul><ul><li>The center and the radius. </li></ul><ul><li>How can you find the center? </li></ul><ul><li>The center is the midpoint of the segment. </li></ul><ul><li>(½ , - ½ ) </li></ul><ul><li>How can you find the radius? </li></ul><ul><li>The radius is the distance from the center to a point on the circle. Use the distance formula. </li></ul><ul><li>The equation is: </li></ul><ul><li> </li></ul>
<ul><li>8. Write in center-radius form and sketch: </li></ul><ul><li> Hint: You must complete the square. </li></ul><ul><li> </li></ul>
<ul><li>What’s the standard form of a line? </li></ul><ul><li>What are the steps for graphing a circle? </li></ul><ul><li>How can you tell if the graph of an equation will be a line, a parabola, or a circle? </li></ul>
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