Which equation represents a circle that contains the point $(-2,8)$ and has a center at $(4,0)$?
Distance formula: $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$
A. $(x-4)^2+y^2=100$
B. $(x-4)^2+y^2=10$
C. $x^2+(y-4)^2=10$
D. $x^2+(y-4)^2=100
Answer (1)
Find the radius r of the circle using the distance formula between the center ( 4 , 0 ) and the point ( − 2 , 8 ) : r = ( − 2 − 4 ) 2 + ( 8 − 0 ) 2 = 10 .
Square the radius to find r 2 = 1 0 2 = 100 .
Substitute the center ( 4 , 0 ) and r 2 into the circle equation ( x − h ) 2 + ( y − k ) 2 = r 2 , which gives ( x − 4 ) 2 + y 2 = 100 .
The equation of the circle is ( x − 4 ) 2 + y 2 = 100 .
Explanation
Problem Analysis The problem asks us to find the equation of a circle given its center and a point it passes through. The general equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . We are given the center ( 4 , 0 ) and a point on the circle ( − 2 , 8 ) .
Calculate the Radius First, we need to find the radius of the circle. The radius is the distance between the center and any point on the circle. We can use the distance formula to find the radius r between the center ( 4 , 0 ) and the point ( − 2 , 8 ) :
r = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 r = ( − 2 − 4 ) 2 + ( 8 − 0 ) 2 r = ( − 6 ) 2 + ( 8 ) 2 r = 36 + 64 r = 100 r = 10 So, the radius of the circle is 10.
Write the Equation of the Circle Now that we have the radius, we can find the equation of the circle. The center is ( 4 , 0 ) , so h = 4 and k = 0 . The radius is r = 10 , so r 2 = 100 . Plugging these values into the circle equation, we get: ( x − h ) 2 + ( y − k ) 2 = r 2 ( x − 4 ) 2 + ( y − 0 ) 2 = 1 0 2 ( x − 4 ) 2 + y 2 = 100 Thus, the equation of the circle is ( x − 4 ) 2 + y 2 = 100 .
Final Answer The equation of the circle that contains the point ( − 2 , 8 ) and has a center at ( 4 , 0 ) is ( x − 4 ) 2 + y 2 = 100 .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, knowing the center and radius helps determine the placement of plants and the overall layout. Similarly, in architecture, circular arches and domes rely on the principles of circle equations to ensure structural integrity and aesthetic appeal. This knowledge also extends to fields like astronomy, where understanding circular orbits is fundamental to predicting the movement of celestial bodies.
