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CIRCLE

CIRCLE. Analytic Geometry Ms. Charmeigne Geil A. Abalos August 23, 2012. Terminology:. A Circle is the set of all points in a plane equidistant from a fixed point. The fixed point is called the center and the positive constant equal distance is called the radius of the circle.

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CIRCLE

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  1. CIRCLE Analytic Geometry Ms. CharmeigneGeil A. Abalos August 23, 2012

  2. Terminology: A Circle is the set of all points in a plane equidistant from a fixed point. The fixed point is called the center and the positive constant equal distance is called the radius of the circle.

  3. Equation of a Circle An equation of the circle can be obtained if its center, C(h,k), and the radius are known.

  4. Theorem 2.28 The circle having center at the point C(h,k) and the radius r has an equation (x-h)2 + (y-k) 2 = r2

  5. Illustration: let P(x,y) be a point on the circle C(h,k) P(x,y)

  6. Then, by Theorem 2.28, we have lPCl = r Hence, (x-h)2 + (y-k) 2 = r Squaring both sides of the equation, we obtain

  7. (x-h)2 + (y-k) 2 = r2 This equation is called the center-radius or the standard form of an equation of a circle with center, C(h,k) and radius r.

  8. Now, if we remove the parentheses of the center-radius form of an equation of the circle, then we obtain x2 – 2hx + h2 + y2 – 2ky + k2 = r2 x2 + y2 – 2hx – 2ky + (h2 + k2 – r2) = 0

  9. Let D=-2h, E=-2k, and F= h2 + k2 – r2. By substitution, the equation becomes x2 + y2+ Dx + Ey + F = 0 This equation is called the general form of an equation of a circle.

  10. Theorem 2.29 Let A, B, C, and D be real numbers such that A=0. Then the graph of the equation Ax2 + Ay2 + Bx + Cy + D = 0 is a circle, a point, or the empty set. This equation can be transformed to the standard form (x-h)2 + (y-k) 2 = q

  11. Remarks: • If q=0, then the only solution to the equation is the point (h,k). Thus, the graph of the equation Ax2 + Ay2 + Bx + Cy + D = 0 is a point. • If q>0, then the equation can be written as (x-h)2 + (y-k) 2 =( q )2. This is the standard form of an equation

  12. of the circle with the center C(h,k) and the radius r= q . Therefore, the graph of the equation Ax2 + Ay2 + Bx + Cy + D = 0 is a circle. (c) If q<0, then (x-h)2 + (y-k) 2 = q has no solution because the sum (x-h)2 + (y-k) 2 is at least zero. Thus, the graph Ax2 + Ay2 + Bx + Cy + D = 0 is the empty set.

  13. Observe the following examples: • Find the general equation of the circle with C(-2, 3) and radius r=4. Solution: using the standard form (x-h)2 + (y-k) 2 = r2 with h=-2, k=3, and r=4, we obtain (x-2)2 + (y-3) 2 = 42 (x2 + 4x + 4) + (y2 – 6y + 9) =16 x2 + y2 + 4x – 6y – 3 = 0 Answer: x2 + y2 + 4x – 6y – 3 = 0

  14. Find the general equation of the circle if its center is C(5,-2) and the circle passes through the point P(-1,5). Solution: r= lCPl= (x1-h)2 + (y1-k)2 r= (-1-5) 2 + (5+2) 2 r= 36+45 r= 85

  15. Using the standard form (x-h)2 + (y-k) 2 = r2 , we have (x-5)2 + (y+2) 2 = ( 85 ) 2 x2 – 10x + 25 + y2 + 4y + 4 = 85 x2 + y2 – 10x + 4y – 56 = 0 Answer: x2 + y2 – 10x + 4y – 56 = 0

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