The equation $(x-1)^2+(y+1)^2=r^2$ represents circle J. The point $D(0,3)$ lies on the circle. What is $r$, the length of the radius of circle J?
A. 17
B. 5
C. $\sqrt{17}$
D. $\sqrt{5}$
Answer (1)
Substitute the coordinates of point D(0,3) into the equation of the circle: ( 0 − 1 ) 2 + ( 3 + 1 ) 2 = r 2 .
Simplify the equation: 1 + 16 = r 2 , which gives r 2 = 17 .
Take the square root of both sides to solve for r : r = 17 .
The radius of circle J is 17 .
Explanation
Problem Analysis We are given the equation of a circle J as ( x − 1 ) 2 + ( y + 1 ) 2 = r 2 , where r is the radius of the circle. We also know that the point D ( 0 , 3 ) lies on the circle. Our goal is to find the value of r .
Substitute the point into the equation Since point D ( 0 , 3 ) lies on the circle, we can substitute its coordinates into the equation of the circle to find r 2 :
( 0 − 1 ) 2 + ( 3 + 1 ) 2 = r 2
Simplify the equation Now, we simplify the equation: ( − 1 ) 2 + ( 4 ) 2 = r 2 1 + 16 = r 2 17 = r 2
Solve for r To find r , we take the square root of both sides of the equation: r = 17 Since the radius must be a positive value, we only consider the positive square root.
Final Answer Therefore, the radius of circle J is 17 .
Examples
Understanding the equation of a circle and how to find its radius given a point on the circle is useful in various real-world applications. For example, when designing a circular garden, you might know the center of the garden and want a specific point on the edge to align with an existing feature. Using the equation of a circle, you can determine the radius needed to achieve this alignment, ensuring your garden design fits perfectly within your landscape.
