Conic Section Circle

Question

The circle is passing through the two points (- 3, 0) and (7,4) and is centered at a point on the y-axis

Answer 1

The center, a point on the y axis, has coordinates (0, y)

As (-3, 0) and (7, 4) are on the circle, so they are the same distance, and the same distance squared, from (0, y)

The distance squared of the center (0, y) to (-3, 0) is

(0 - -3)² + (y - 0)² = 9 + y²

The distance squared of the center (0, y) to (7, 4) is

(0 - 7)² + (y - 4)² = 49 + y² - 8y + 16 = y² - 8y + 65

The distance squared is equal for the two, so

y² - 8y + 65 = 9 + y²

56 = 8y

y = 7

The center is at (0, 7)

The distance squared is 9 + y² = 58

(x - 0)² + (y - 7)² = 58

x² + y² - 14y + 49 = 58

x² + y² - 14y = 9, or

x² + y² - 14y - 9 = 0

Answer 2

Here is a faster solution to the problem.

Let A(- 3, 0) and B(7,4)

Now there is an infinitude of circles that are passing through A and B. But since its given that the center of

the circle, let us call it K, lies on the y-axis its coordinates then the coordinates of the center K are the the coordinates of the point where

the perpendicular bisector of the line segment AB intersects the y-axis

The perpendicular bisector of the line segment AB has equation y = -(5/2) x +7.

Hence the equation of the circle is x² + (y - 7)² = 58 since the KA= √58