Find the equation of the circle that passes through the point F (5, -2) and the center which is at the point C (-1,1)

First, let’s find what the radius of a given circle is equal to.

In the initial data for this task, it is reported that this circle centered at point C (-1; 1) passes through point F (5; -2), therefore, the radius of this circle is equal to the length of the segment CF and is:

√ (5 – (-1)) ^ 2 + (-2 – 1) ^ 2) = √ (5 + 1) ^ 2 + (-2 – 1) ^ 2) = √ (6 ^ 2 + 3 ^ 2 ) = √ (36 + 9) = √45.

Knowing the radius of the circle and its center, we can write down the equation of the given circle:

(x – (-1)) ^ 2 + (y – 1) ^ 2 = (√45) ^ 2,

simplifying which, we get:

(x + 1) ^ 2 + (y – 1) ^ 2 = 45.

Answer: (x + 1) ^ 2 + (y – 1) ^ 2 = 45.