Calculus Problem

Question

Find an equation of the circle whose center is at (2, -1) and that passes through (4,5).

Answer 1

The equation of a circle is:

(x-h)² + (y-k)² = r²

where h and k are the circle's center coordinates and r is the radius. From the problem, we are given the center and only need to calculate the radius.

This can be accomplished by calculating the radius given the point the circle passes through and the center using Pythagorean's Theorem where the radius is the hypotenuse and the legs of the right triangle are the horizontal and vertical distance between the center and the point.

The horizontal distance is the distance along the x-axis and is 2 units. This is because we begin at the circle with its center point's X-value of 2 and moving two units along the x-axis to arrive at the point's x-value of 4.

The vertical distance is the distance along the y-axis and is 6 units. This is because we begin at -1 along the y-axis and move up 6 units to arrive at the point's y-value of 5.

Given these, we solve for r²:

r² = (x)² + (y)² = 2² + 6² = 4 + 36 = 40

r² = 40.

Now that we have the value for r², we substitute this and the values of the circle's center into the circle equation:

(x-2)² + (y-(-1))² = 40

We will simply the y section:

(x-2)² + (y+1)² = 40

Now we have the equation of the circle.