Question

3 which equation represents a circle centered at (3,5) and passing through the point (-2,9)? a. (x + 3)² + (y + 5)² = 41 b. (x − 3)² + (y − 5)² = 41 c. (x + 3)² + (y + 5)² = 17 d. (x − 3)² + (y − 5)² = 17

Explanation:

Step1: Recall the circle equation formula

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. Here, the center \((h, k)=(3, 5)\), so the equation starts as \((x - 3)^2 + (y - 5)^2 = r^2\). This eliminates options A and C.

Step2: Calculate the radius squared

To find \(r^2\), use the distance formula between the center \((3, 5)\) and the point \((-2, 9)\) on the circle. The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), so \(r^2=(x_2 - x_1)^2+(y_2 - y_1)^2\). Substituting \(x_1 = 3\), \(y_1 = 5\), \(x_2=-2\), \(y_2 = 9\):
\[

$$\begin{align*} r^2&=(-2 - 3)^2+(9 - 5)^2\\ &=(-5)^2+4^2\\ &=25 + 16\\ &=41 \end{align*}$$

\]

Answer:

B. \((x - 3)^2 + (y - 5)^2 = 41\)