Question
A circle has a diameter with endpoints at (5, −8) and (13, −16). What is the value of k in the circle's equation, (x−h)2+(y−k)2=r2?(1 point)
Answers
GPT-4o mini
To find the value of \( k \) in the circle's equation \((x-h)^2 + (y-k)^2 = r^2\), we first need to determine the center of the circle, which is located at the midpoint of the diameter.
The endpoints of the diameter are given as \((5, -8)\) and \((13, -16)\). The formula for the midpoint \((h, k)\) of two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2}
\]
Substituting in our points:
\[
h = \frac{5 + 13}{2} = \frac{18}{2} = 9
\]
\[
k = \frac{-8 + (-16)}{2} = \frac{-24}{2} = -12
\]
Thus, the values of \( h \) and \( k \) are \( 9 \) and \( -12 \), respectively.
The value of \( k \) is \(\boxed{-12}\).
The endpoints of the diameter are given as \((5, -8)\) and \((13, -16)\). The formula for the midpoint \((h, k)\) of two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2}
\]
Substituting in our points:
\[
h = \frac{5 + 13}{2} = \frac{18}{2} = 9
\]
\[
k = \frac{-8 + (-16)}{2} = \frac{-24}{2} = -12
\]
Thus, the values of \( h \) and \( k \) are \( 9 \) and \( -12 \), respectively.
The value of \( k \) is \(\boxed{-12}\).