Question

which equation represents a circle that contains the point $(-5,-3)$ and has a center at $(-2,1)$?
distance formula: $sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$
$(x + 2)^{2}+(y - 1)^{2}=5$
$(x + 2)^{2}+(y - 1)^{2}=25$
$(x - 1)^{2}+(y + 2)^{2}=5$
$(x - 1)^{2}+(y + 2)^{2}=25$

Explanation:

Step1: Find the radius

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \((x_1,y_1)=(-2,1)\) (center) and \((x_2,y_2)=(-5,-3)\) (point on the circle).

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Step2: Write the equation of the circle

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. Here, \(h=-2,k = 1,r = 5\).

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Answer:

\((x + 2)^2+(y - 1)^2=25\) (the second option)