Select the correct answer.
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 ( x − 1 ) 2 + ( y + 1 ) 2 = r 2 .
Simplify the equation to find r 2 = 17 .
Take the square root to solve for r .
The radius of circle J is 17 .
Explanation
Problem Analysis The equation of circle J is given by ( x − 1 ) 2 + ( y + 1 ) 2 = r 2 . We know that point D ( 0 , 3 ) lies on the circle. To find the radius r , we substitute the coordinates of point D into the equation of the circle.
Substitution Substitute x = 0 and y = 3 into the circle equation: ( 0 − 1 ) 2 + ( 3 + 1 ) 2 = r 2
Simplification Simplify the equation: ( − 1 ) 2 + ( 4 ) 2 = r 2 1 + 16 = r 2 17 = r 2
Solving for r Solve for r by taking the square root of both sides: r = 17
Final Answer Therefore, the radius of circle J is 17 .
Examples
Understanding the radius of a circle is crucial in many real-world applications. For instance, when designing a circular garden, knowing the radius helps determine the amount of fencing needed. Similarly, in architecture, calculating the radius of a dome is essential for structural stability and material estimation. The equation of a circle and its radius are fundamental concepts that bridge theoretical math with practical design and construction.
