Question

select the correct answer. which equation represents circle n? image: circle with center ( n(-7,6) ) and point ( l(-2,11) ) on the circle options: 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 circle equation

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

Step2: Identify the center \((h,k)\)

The center \(N\) is \((-7,6)\), so \(h = -7\) and \(k = 6\). Substituting into the standard form, we get \((x - (-7))^2 + (y - 6)^2 = r^2\), which simplifies to \((x + 7)^2 + (y - 6)^2 = r^2\). This eliminates options B and D.

Step3: Calculate the radius \(r\)

Use the distance formula between center \(N(-7,6)\) and point \(L(-2,11)\). The distance formula is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substituting \(x_1 = -7\), \(y_1 = 6\), \(x_2 = -2\), \(y_2 = 11\), we get:
\[

$$\begin{align*} r&=\sqrt{(-2 - (-7))^2 + (11 - 6)^2}\\ &=\sqrt{(5)^2 + (5)^2}\\ &=\sqrt{25 + 25}\\ &=\sqrt{50} \end{align*}$$

\]
Then \(r^2 = (\sqrt{50})^2 = 50\).

Step4: Form the circle's equation

Substituting \(r^2 = 50\) and the center into the equation, we get \((x + 7)^2 + (y - 6)^2 = 50\).

Answer:

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