What is the equation of the circle with center (2, –5) that passes through the point (–2, 10)?

(x – 2)2 + (y – (–5))2 = 25
(x – (–2))2 + (y – 10)2 = 241
(x – 2)2 + (y – (–5))2 = 241
(x – (–2))2 + (y – 10)2 = 25

I'm glad you asked this question! The equation of the circle with center (2, -5) that passes through the point (-2, 10) is (x - 2)^2 + (y - (-5))^2 = 241. It's like magic, but with numbers!

To find the equation of the circle with center (2, –5) that passes through the point (–2, 10), we can use the formula for the equation of a circle:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) represents the center of the circle, and r represents the radius.

Substituting the given values into the formula:

(x - 2)^2 + (y - (-5))^2 = r^2

Simplifying, we get:

(x - 2)^2 + (y + 5)^2 = r^2

To find the radius squared, we can use the distance formula between the center (2, -5) and the point (-2, 10):

r^2 = (-2 -2)^2 + (10 - (-5))^2
= (-4)^2 + (15)^2
= 16 + 225
= 241

So the equation of the circle is:

(x - 2)^2 + (y + 5)^2 = 241

Therefore, the correct answer is:

(x - 2)^2 + (y + 5)^2 = 241

The equation of a circle with center (h, k) and radius r is given by (x – h)^2 + (y – k)^2 = r^2.

To find the equation of the circle with center (2, -5) that passes through the point (-2, 10), we need to determine the radius of the circle.

The radius is the distance between the center and any point on the circle. In this case, we can use the distance formula to find the radius.

The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the distance between the center (2, -5) and the given point (-2, 10):

d = sqrt((-2 - 2)^2 + (10 - (-5))^2)
d = sqrt((-4)^2 + (15)^2)
d = sqrt(16 + 225)
d = sqrt(241)

So, the radius of the circle is sqrt(241).

Substituting the center (2, -5) and the radius sqrt(241) into the general equation of a circle, we get:

(x – 2)^2 + (y – (-5))^2 = (sqrt(241))^2
(x – 2)^2 + (y + 5)^2 = 241
(x – 2)^2 + (y + 5)^2 = 241

Therefore, the correct equation of the circle is (x – 2)^2 + (y + 5)^2 = 241.
So, none of the given options are correct.

You know that the choice must be either #1 or #3, since they have the proper center.

(-2,10) fits only choice #3.