Question

1. which of the following is an equation of a circle in the xy - plane with center (-2,3) and a radius with endpoint (5,1)?

(a) (x + 2)^2+(y - 3)^2 = 53
(b) (x - 2)^2+(y + 3)^2 = 53
(c) (x - 2)^2+(y + 3)^2 = \sqrt{53}
(d) (x + 2)^2+(y - 3)^2 = \sqrt{53}

Explanation:

Step1: Find the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \((x_1,y_1)=(-2,3)\) and \((x_2,y_2)=(5,1)\).
\(r=\sqrt{(5 + 2)^2+(1 - 3)^2}=\sqrt{49+4}=\sqrt{53}\). Then \(r^{2}=53\).

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. Given \((h,k)=(-2,3)\) and \(r^{2}=53\), the equation is \((x + 2)^2+(y - 3)^2=53\).

Answer:

A. \((x + 2)^2+(y - 3)^2 = 53\)