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Find the chord length

Question:

A line passes the origin and has the inclined angle 120o. This line intersects a circle at two points. The circle has the equation x2 + y2 - 6y = 0. What is the chord length cut by this line?

Solution:

The line passes the origin, so the line equation is y = k x, in which k is the slope of the line. Because the line has an inclined angle 120o, so, the slope k is:

k = tan 120o
= tan(180o - 60o)
= - tan60o
= - square root of 3.

So, the line equation is: y = - (square root of 3) x. Now we change this line equation to the general form of a line equation.

The general form of the given line equation is: (square root of 3) x + y = 0. The standard line equation is: Ax + By + C = 0. Compare the standard line equation with the given line equation, we get A = square root of 3, B = 1 and C = 0.

Now we will find the center of the circle and the radius of the circle.

x2 + y2 - 6y = 0
x2 + y2 - 6y + 32 - 32 = 0
x2 + (y - 3)2 = 32

The standard circle equation is: (x - xo)2 + (y - yo)2 = r2, in which xo is the x-coordinate of the center of the circle, yo is the y-coordinate of the center of the circle and r is the radius of the circle. Compare the standard circle equation with the given circle equation, we get, xo = 0, yo = 3, and r = 3.

The center point of the circle is O1, its coordinate is O1(0, 3). Because the radius of the circle is 3, so, the circle passes the origin O(0, 0).

what is the chord length when a line passes the origin and has an inclined angle 120 degrees intersects a circle?

Now we find the distance from the center of the circle O1(0, 3) to the line y = -(square root of 3)x.

We draw a line from the center of the circle O1 perpendicular to the given line, y = -(square root of 3) x. This line intersects the given line at point Q. Q is the perpendicular point. The d is distance from the center of the circle to the given line y = -(square root of 3)x.

d = |O1Q| = |Axo + Byo + C|/square root of (A2 + B2)
= |1 × 3|/square root of [(square root of 3)2 + 12]
= 3/square root of 4
= 3/2

The given line intersects the circle at two points, one point is the origin O(0, 0) and the other point is E.

Now we find the chord length |EO|. we use the formula: the chord length |EO| = 2 square root of (r2 - d2), in which, r is the radius of the circle, d is the distance from the center of the circle to the line. E is the point that the given line intersects the circle.

|EO| = 2 square root of (r2 -d2)
= 2 square root of [32 - (3/2)2]
= 2 square root of (9 - 9/4)
= 2 square root of (36/4 - 9/4)
= 2 square root of 27/4
= 2 × (1/2) × 3 × square root of 3
= 3 square root of 3

The chord length is 3 square roots of 3. Watch the video for more details.