if p=(1,4) and q=(7,-2), find the equation of the circle with \overline{pq} as its diameter. (x - ?)^2 + (y - \square)^2 = \square

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Accepted Answer

# Explanation:
## Step1: Find the center of the circle
The center of the circle (mid - point of the diameter) with endpoints $P(x_1,y_1)=(1,4)$ and $Q(x_2,y_2)=(7, - 2)$ is given by the mid - point formula $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.
$$
\begin{align*}
h&=\frac{1 + 7}{2}=\frac{8}{2}=4\
k&=\frac{4+( - 2)}{2}=\frac{2}{2}=1
\end{align*}
$$
## Step2: Find the radius of the circle
The radius $r$ is half the length of the diameter. The length of the diameter $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ (distance formula).
$$
\begin{align*}
d&=\sqrt{(7 - 1)^2+( - 2 - 4)^2}\
&=\sqrt{6^2+( - 6)^2}\
&=\sqrt{36 + 36}\
&=\sqrt{72}=6\sqrt{2}
\end{align*}
$$
$r=\frac{d}{2}=\frac{6\sqrt{2}}{2}=3\sqrt{2}$, and $r^{2}=(3\sqrt{2})^{2}=18$

# Answer:
$(x - 4)^{2}+(y - 1)^{2}=18$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the center of the circle

Step2: Find the radius of the circle

Answer: