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Parabolas and Circles Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Equation of a Circle

The general form for the equation of a circle in the xy -plane is given by the following formula:

( xx 0 ) 2 + ( yy 0 ) 2 = r 2

where ( x 0 , y 0 ) represents the coordinates of the center of the circle, and r represents the radius . This is illustrated in Fig. 6-9. In the special case where the circle is centered at the origin, the formula becomes:

x 2 + y 2 = r 2

 

 

The Cartesian Plane Parabolas and Circles Equation Of Circle

Fig. 6-9 . Circle centered at ( x 0 , y 0 ) with radius r .

Such a circle intersects the x axis at the points ( r , 0) and (− r , 0); it intersects the y axis at the points (0, r ) and (0, − r ). An even more specific case is the unit circle:

x 2 + y 2 = 1

This curve intersects the x axis at the points (1,0) and (−1, 0); it intersects the y axis at the points (0,1) and (0, −1).

Parabolas and Circles Practice Problems

PROBLEM 1

Draw a graph of the circle represented by the equation ( x − 1) 2 + ( y + 2) 2 = 9.

SOLUTION 1

zBased on the general formula for a circle, we can determine that the center point has coordinates x 0 = 1 and y 0 = −2. The radius is equal to the square root of 9, which is 3. The result is a circle whose center point is (1,−2) and whose radius is 3. This is shown in Fig. 6-10.

The Cartesian Plane Parabolas and Circles Equation Of Circle

Fig. 6-10 . Illustration for Problem 6-3.

PROBLEM 2

Determine the equation of the circle shown in Fig. 6-11.

The Cartesian Plane Parabolas and Circles Equation Of Circle

Fig. 6-11 . Illustration for Problem 6-4.

SOLUTION 2

First note that the center point is (−8, −7). That means x 0 = −8 and y 0 = −7. The radius, r , is equal to 20, so r 2 = 20 × 20 = 400. Plugging these numbers into the general formula gives us the equation of the circle, as follows:

( xx 0 ) 2 + ( yy 0 ) 2 = r 2

              [ x − (−8)] 2 + [ y − (−7)] 2 = 400

   ( x + 8) 2 + ( y + 7) 2 = 400

Practice problems for these concepts can be found at: The Cartesian Plane Practice Test.

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