Question

question 14 of 22 which of the following is the equation of the circle in the xy - plane with center at point (0, 5) and a radius with endpoint (-3, 9)? a. (x + 3)^2+(y - 9)^2 = 25 b. x^2+(y - 5)^2 = 5 c. x^2+(y - 5)^2 = 25 d. x^2+(y - 5)^2 = 205

Explanation:

Step1: Recall circle - equation formula

The standard form of the equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). The center of the circle is \((0,5)\) and a point on the circle is \((- 3,9)\).

Step2: Calculate the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the radius. Here, \((x_1,y_1)=(0,5)\) and \((x_2,y_2)=(-3,9)\). Then \(r=\sqrt{(-3 - 0)^2+(9 - 5)^2}=\sqrt{(-3)^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step3: Write the equation of the circle

Substitute \(h = 0\), \(k = 5\), and \(r = 5\) into the standard - form equation \((x - h)^2+(y - k)^2=r^2\). We get \(x^{2}+(y - 5)^{2}=25\).

Answer:

C. \(x^{2}+(y - 5)^{2}=25\)