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Writing the equation of a circle

Writing the equation of a circle. A unit circle is a circle with radius of 1, centered at the origin. Analyzing a circle centered at the origin. Suppose P(x,y) is a point on the circle with a 1-unit radius that is centered on the origin. What is QP? How is QP related to x and y?. P. y. Q.

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Writing the equation of a circle

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  1. Writing the equation of a circle A unit circle is a circle with radius of 1, centered at the origin

  2. Analyzing a circle centered at the origin • Suppose P(x,y) is a point on the circle with a 1-unit radius that is centered on the origin. What is QP? How is QP related to x and y? P y Q x

  3. solution • Since QP is a radius, QP = 1. In the diagram, tri PQR is a right triangle, so by Pythagorean theorem, x2 + y2 = 1 • Any circle centered at the origin has an equation x2 + y2 = r2, where r is the radius

  4. Equation of circle not centered on the origin(x-h)2 + (y-k)2 = r2, where (h,k) is the center of the circle • Suppose the circle is centered at (2,0) with a radius of 2. The distance from the center to the circle is 2 • So • (x-2)2 + (y-0)2 = 22

  5. Writing the equation for a circle from a graph • Center is at ? Radius is ? . D

  6. Concentric circles • Two circles are concentric if they have the same center

  7. Circle E is concentric with circle D • Circle D has a radius of 3 and circle E has a radius of 5. The center of both circles is (1,3) • Find the equation of circle D • (x-1)2 +(y-3) 2 = 32 • Find the equation of circle E. • (x-1)2 + (y-3)2 = 52

  8. Graphing a circle given its equation • Graph x2+y2= 25 • 1) check for h and k • Since they are both 0, the center is at the origin • 2) find the radius • 52 = 25,so radius is 5 • 3) find endpoints of the vertical and horizontal radii • (5,0) (0,5) (-5,0) (0,-5) • Using these as a guide, draw the circle

  9. Graph (x-2)2 + (y+1)2 = 16 • Step 1-find h and k, then locate origin • Step 2- find radius, then locate the endpoints of the vertical and horizontal radii. • Step 3- graph

  10. practice • 1. If M(x,y) is a point centered at the origin with a radius of 3, what is PM and what is the equation of the circle? • 2. Write an equation of a circle with a radius of square root of 2, which is centered at the origin. • 3. write the equation of a circle with center (2,0) and endpoints of vertical and horizontal radii at (8,0) (-4,0) (2,4) (2, -4) • 4. Circle D is concentric with the circle in problem 3 and has a radius of 3.5.Write the equationof circle D.

  11. practice • 5. The equation of circle D is x2 +y2 = 6.25 • Graph it! • 6. The equation of circle E is • (x+1)2 + (y-3)2= 4. Graph it!

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