Conic Section: Circles


 Milo Barnett
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1 Conic Section: Circles Circle, Center, Radius A circle is defined as the set of all points that are the same distance awa from a specific point called the center of the circle. Note that the circle consists of onl the curve but not of the area inside the curve. The distance from the center to the circle is called the radius of the circle. We often label the center with a capital letter and we refer to the circle b that letter. For eample, the circle at the right is called circle A or A. A center radius Congruent Circles Two circles are congruent if the have the same radius, regardless of where their centers are located. For eample, all circles of radius of centimeters are congruent to each other. Similarl, all circles with a radius of 5 miles are congruent to each other. If circles are not congruent, then the are similar with the similarit ratio given b the ratio of their radii. Chord, Diameter, Secant A chord is defined as a line segment starting at one point on the circle and ending at another point on the circle. A chord that goes through the center of the circle is called the diameter of the circle. Notice that the diameter is twice as long as the radius of the circle. A secant is a line that cuts through the circle and continues infinitel in both directions. tangent Point of Tangenc and Tangent A tangent line is defined as a line that touches the circle at eactl one point. This point is called the point of tangenc. point of tangenc TangentRadius Theorem The radius is perpendicular to the tangent line at the point of tangenc.
2 Equations and Graphs of Circles A circle is defined as the set of all points that are equidistant from a single point called the center. This definition can be used to find an equation of a circle in the coordinate plane. Let s consider the circle shown at the right. As ou can see this circle has its center at point (, ) and it has a radius of 3. All the points (, ) on the circle are a distance of 3 units awa from the center of the circle. (, ) 3 (, ) We can epress this information as an equation with the help of the Pthagorean Theorem. The right triangle shown in the figure has legs of length and and hpotenuse of length 3. We write: ( ) ( ) We can generalize this equation for a circle with center at point (h, k) and radius r. ( h) ( k) r Eample: Find the center and radius of the following circles: a) ( ) ( 1) 5 b) ( 1) ( ) a) We rewrite the equation as: ( ) ( 1) 5 The center of the circle is at point (, 1) and the radius is 5. b) We rewrite the equation as: ( ( 1)) ( ) The center of the circle is at point (1, ) and the radius is. Eample: Graph the following circles: a) b) ( ) 1
3 In order to graph a circle, we first graph the center point and then draw points that are the length of the radius awa from the center. a) We rewrite the equation as: ( 0) ( 0) 3 The center of the circle is point at (0, 0) and the radius is 3. b) We rewrite the equation as: ( ( )) ( 0) 1 The center of the circle is point at (, 0) and the radius is 1. Eample 5: Write the equation of the circle in the graph. From the graph we can see that the center of the circle is at point (, ) and the radius is 3 units long. Thus the equation is: ( ) ( )
4 Eample: Determine if the point (1, 3) is on the circle given b the equation: ( 1) ( 1) 1 In order to find the answer, we simpl plug the point (1, 3) into the equation of the circle. ( 1 1) ( 3 1) The point (1, 3) satisfies the equation of the circle. Eample: Find the equation of the circle whose diameter etends between points A = (3, ) and B = ( 1, ). The general equation of a circle is: ( h) ( k) r In order to write the equation of the circle in this eample, we need to find the center of the circle and the radius of the circle. Let s graph the two points on the coordinate plane. We see that the center of the circle must be in the middle of the B diameter. In other words, the center point is midwa between the two points (1,1) A and B. To get from point A to point B, we must travel units to the right and units up. To get halfwa from point A to point B, we must travel units to the right and 3 units up. This means the A center of the circle is at point (3 +,  + 3) or (1, 1). We find the length of the radius using the Pthagorean Theorem: r 3 r 13 r 13 Thus, the equation of the circle is: ( 1) ( 1) 13 Completing the Square: You saw that the equation of a circle with center at point ( 0, 0 ) and radius r is given b: ( h) ( k) r This is called the standard form of the circle equation. The standard form is ver useful because it tells us right awa what the center and the radius of the circle is. If the equation of the circle is not in standard form, we use the method of completing the square to rewrite the equation in the standard form.
5 Eample: Find the center and radius of the following circle and sketch a graph of the circle. 0 To find the center and radius of the circle we need to rewrite the equation in standard form. The standard equation has two perfect square factors one for the terms and one for the terms. We need to complete the square for the terms and the terms separatel. To complete the squares we need to find which constants allow us to factors each trinomial into a perfect square. To complete the square for the terms we need to add a constant of on both sides. To complete the square for the terms we need to add a constant of on both sides. We can factors the separate trinomials and obtain: 13 3 ) ( ) ( This simplifies as: 3 ) ( ) ( You can easil see now that the center of the circle is at point (, 3) and the radius is.
6 Eercises: Find the center and the radius of the following circles: ( 3) ( 5) ( ). ( ) ( 1) 5 7. Check that the point (3, ) is on the circle given b the equation ( ). 8. Check that the point (5, 5) is on the circle given b the equation ( 3) ( ) 50.. Write the equation of the circle with center at (, 0) and radius. 10. Write the equation of the circle with center at (, 5) and radius. 11. Write the equation of the circle with center at (1, 5) and radius 10. Write the equation of the following circles:
7 1. In a circle with center (, 1) one endpoint of a diameter is (1, 3). Find the other endpoint of the diameter. 15. The endpoints of the diameter of a circle are given b the points A = (1, ) and B = (7, ). Find the equation of the circle. 1. A circle has center (0, ) and contains point (1, 1). Find the equation of the circle. 17. A circle has center (, ) and contains point (, ). Find the equation of the circle. 18. Find the center and the radius of the following circle: Find the center and the radius of the following circle: Find the center and the radius of the following circle:
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