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}}\\)
\\(\bigcirc\\) \\((x - 1)^{2}+(y + 2)^{2}=25\\)
\\(\bigcirc\\) \\((x + 2)^{2}+(y - 1)^{2}=5\\)
\\(\bigcirc\\) \\((x + 2)^{2}+(y - 1)^{2}=25\\)
\\(\bigcirc\\) \\((x - 1)^{2}+(y + 2)^{2}=5\\)

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).

$$
LATEXBLOCK0
$$

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. Here, \(h=-2,k = 1,r = 5\).

$$
LATEXBLOCK1
$$

Answer:

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