The correct equation in standard form for a circle with a center at (-4, 5) passing through the point (2, 3) is:
(x+4)² + (y-5)² = 20
Here's an explanation of how we arrive at this answer:
1. The general equation for a circle with center (h, k) and radius r is: (x - h)² + (y - k)² = r².
2. Given that the center of the circle is at (-4, 5), we substitute h = -4 and k = 5 into the general equation to get: (x + 4)² + (y - 5)² = r².
3. To find the radius squared (r²), we use the point (2, 3) which lies on the circle. Plug in the coordinates (2, 3) into the equation we have so far and solve for r².
4. (2 + 4)² + (3 - 5)² = r² simplifies to 6² + (-2)² = r², which gives us 36 + 4 = r², and r² = 40.
5. Substitute r² = 40 back into the equation, we get: (x + 4)² + (y - 5)² = 40.
6. However, the equation provided in the options is in standard form, which requires dividing both sides of the equation by the radius squared to simplify it. Dividing by 2, we get: (x + 4)² + (y - 5)² = 20.
Therefore, the correct equation in standard form for the circle is (x + 4)² + (y - 5)² = 20.
