Question

find the equation for the circle with center (4,2) and passing through (-4,-5).

write the equation for the circle.

(simplify your answer)

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. Here, \(h = 4\), \(k=2\).

Step2: Calculate the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the radius. The center \((x_1,y_1)=(4,2)\) and the point on the circle \((x_2,y_2)=(-4,-5)\).

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Step3: Substitute into the circle's equation

Substitute \(h = 4\), \(k = 2\), and \(r=\sqrt{113}\) into \((x - h)^2+(y - k)^2=r^2\).

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Answer:

\((x - 4)^2+(y - 2)^2=113\)