Select the correct answer.
Circle $C$ has a center at $(-2,10)$ and contains the point $P(10,5)$. Which equation represents circle $C$ ?
A. $(x-2)^2+(y+10)^2=13$
B. $(x-2)^2+(y+10)^2=169$
C. $(x+2)^2+(y-10)^2=13$
D. $(x+2)^2+(y-10)^2=169
Answer (1)
Find the radius of the circle using the distance formula: r = ( 10 − ( − 2 ) ) 2 + ( 5 − 10 ) 2 = 13 .
Square the radius to find r 2 = 1 3 2 = 169 .
Substitute the center ( − 2 , 10 ) and r 2 into the circle equation: ( x − ( − 2 ) ) 2 + ( y − 10 ) 2 = 169 .
The equation of circle C is ( x + 2 ) 2 + ( y − 10 ) 2 = 169 .
Explanation
Problem Analysis The problem provides the center of a circle C at ( − 2 , 10 ) and a point P ( 10 , 5 ) on the circle. We need to find the equation of circle C . The general equation of a circle with center ( h , k ) and radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 .
Calculate the radius First, we need to find the radius r of the circle. The radius is the distance between the center ( − 2 , 10 ) and the point P ( 10 , 5 ) on the circle. We can use the distance formula to find the radius:
r = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Substituting the coordinates of the center and the point P :
r = ( 10 − ( − 2 ) ) 2 + ( 5 − 10 ) 2 r = ( 10 + 2 ) 2 + ( 5 − 10 ) 2 r = ( 12 ) 2 + ( − 5 ) 2 r = 144 + 25 r = 169 r = 13
Write the equation of the circle Now that we have the radius r = 13 , we can find r 2 :
r 2 = 1 3 2 = 169
The equation of the circle with center ( − 2 , 10 ) and radius 13 is: ( x − ( − 2 ) ) 2 + ( y − 10 ) 2 = 169 ( x + 2 ) 2 + ( y − 10 ) 2 = 169
Final Answer Comparing this equation with the given options, we see that it matches option D. Therefore, the correct equation representing circle C is ( x + 2 ) 2 + ( y − 10 ) 2 = 169 .
Examples
Understanding circle equations is crucial in various fields, such as computer graphics, where circles are frequently used to create shapes and designs. For instance, in a video game, you might need to draw a target with a circular shape. Knowing the center and radius, you can define the circle's equation and use it to render the target accurately on the screen. This ensures that the game's visuals are precise and appealing, enhancing the player's experience.
