Question

find the center and the radius of the circle with the given equation. then draw the graph.
$x^{2}+y^{2}-10x + 2y = 38$

use the graphing tool to graph the circle.
click to enlarge graph

Explanation:

Step1: Complete the square for x

The equation is \(x^{2}+y^{2}-10x + 2y=38\). For the \(x\)-terms: \(x^{2}-10x\), take half of \(- 10\) which is \(-5\), square it: \((-5)^{2}=25\). Add and subtract 25 (but we will add to both sides).

Step2: Complete the square for y

For the \(y\)-terms: \(y^{2}+2y\), take half of \(2\) which is \(1\), square it: \(1^{2} = 1\). Add and subtract 1 (but we will add to both sides).

Step3: Rewrite the equation

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The standard form of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.

Answer:

The center of the circle is \((5,-1)\) and the radius is \(8\) (since \(r^{2}=64\), so \(r = \sqrt{64}=8\)).