Question

which equation represents circle n?
a. $(x + 7)^2+(y - 6)^2 = 25$
b. $(x - 7)^2+(y + 6)^2 = 25$
c. $(x + 7)^2+(y - 6)^2 = 50$
d. $(x - 7)^2+(y + 6)^2 = 50$

Explanation:

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. Here, the center \(N(-7,6)\), so \(h=-7\), \(k = 6\). The equation becomes \((x+7)^2+(y - 6)^2=r^2\).

Step2: Calculate the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the radius \(r\) (distance between \(N(-7,6)\) and \(L(-2,11)\)).
\(r=\sqrt{(-2+7)^2+(11 - 6)^2}=\sqrt{(5)^2+(5)^2}=\sqrt{25 + 25}=\sqrt{50}\). Then \(r^2 = 50\).

Answer:

C. \((x + 7)^2+(y - 6)^2=50\)