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Circle and line example

Question:

A line passes the origin and intersects a circle x2 + 4x + y2 - 6y + 9 = 0. The chord that the line intersecting the circle is 4. Find the equation of the line.

Solution:

First, we need to find the center of the circle and the radius of the circle.

x2 + 4x + y2 - 6y + 9 = 0
x2 + 4x + 4 - 4 + y2 - 6y + 9 = 0
(x + 2)2 + (y - 3)2 = 4
(x + 2)2 + (y - 3)2 = 22

The standard circle equation is: (x - x0)2 + (y - y0)2 = r2, in which the center of the circle is (x0 , y0) and the radius of the circle is r. Compare the standard circle equation with the given circle equation, we get the center of the circle is (-2, 3) and the radius of the circle is 2.

Because the line passes the origin, so, the line equation is: y = kx, in which k is the slope of the line.

When the line intersects the circle and generate a chord, the line intersects the circle at two points. The line segment inside the circle including the two endpoint is the chord. Only when the line passes the center of the circle, the chord that passes the center of the circle is the diameter of the circle. Because the radius of the circle is 2, so the diameter of the circle is 4. Because we are given that the chord is 4, so, the line must pass the center of the circle.

Because the point (-2, 3) lies on the line, so the point (-2, 3) satisfies the line equation. Now substitute the point (-2, 3) into the line equation to get the value of k.

y = kx
3 = k (-2)
k = -(3/2)

Therefore, the line equation is y = - (3/2) x. Watch the video for more details.