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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

CIRCLE

Analytic Geometry

Ms. CharmeigneGeil A. Abalos

August 23, 2012


Terminology
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.


Equation of a circle
Equation of a Circle

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


Theorem 2 28
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


Illustration let p x y be a point on the circle
Illustration: let P(x,y) be a point on the circle

C(h,k)

P(x,y)


Then by theorem 2 28 we have
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


X h 2 y k 2 r 2
(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.


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


Let D=-2 of an equation of the circle, then we obtainh, 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.


Theorem 2 29
Theorem 2.29 of an equation of the circle, then we obtain

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


Remarks
Remarks: of an equation of the circle, then we obtain

  • 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


of the circle with the center C( of an equation of the circle, then we obtainh,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.


Observe the following examples
Observe the following examples: of an equation of the circle, then we obtain

  • 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



Using the standard form (x- C(5,-2) and the circle passes through the point P(-1,5).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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