Question

Find the equation of the circle with center at (-1,4) and passes through the point (-2,2)?

a) (x + 1)^2 + (y - 4)^2 = 10
b) (x - 1)^2 + (y + 4)^2 = 10
c) (x + 1)^2 + (y - 4)^2 = 5
d) (x - 1)^2 + (y + 4)^2 = 5

Answer 1

The equation of the circle with center (-1,4) that passes through the point (-2,2) is found using the distance formula to calculate the radius, which is √5. Thus, the equation is (x + 1)² + (y - 4)² = 5, which corresponds to option (c).

Step-by-step explanation:

To find the equation of a circle with a center at (-1,4) and that passes through the point (-2,2), we first need to calculate the radius of the circle. The radius is the distance between the center of the circle and any point on the circle. We use the distance formula to find the radius, r:

r = √[(-2 - (-1))² + (2 - 4)²] = √[(-1)² + (-2)²] = √[1 + 4] = √5

Now we have the radius, and we can write down the equation of the circle. A circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle, and r is the radius. Substituting the center (-1,4) and the radius √5, we get:

(x - (-1))² + (y - 4)² = (√5)²

This simplifies to:

(x + 1)² + (y - 4)² = 5

Matching this with the options provided, the correct answer is (c) (x + 1)² + (y - 4)² = 5.