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Hyperbola and circle example

Question:

A hyperbola has the equation x2/16 - y2/9 = 1. The left focus of the hyperbola is the center of a circle. The circle is tangent to the asymptotes of the hyperbola. What is the equation of the circle?

Solution:

The standard hyperbola equation is: x2/a2 - y2/b2 = 1 (a > 0, b > 0). Compare the given hyperbola equation with the standard hyperbola equation, we get, a = 4 and b= 3.

The coordinate of the left focus of the hyperbola is F1(-c, 0), in which c2 = a2 + b2 = 42 + 32 = 16 + 9 = 25. So, c = 5. The coordinate of the left focus of the hyperbola is F1(-5, 0).

Because the center of the circle is the left focus of the hyperbola, so the coordinate of the center of the circle is (-5, 0).

We are given that the circle is tangent to the asymptotes of the hyperbola. The equation of the asymptotes of a hyperbola is: y = +- (b/a) x. So, one asymptote is y = (3/4) x and other asymptotes is y = -(3/4) x.

Now we draw a line from the center of the circle, F1 (-5, 0), perpendicular to the asymptotes, y = -(3/4) x. This line intersects the asymptotes at point Q.

Because the circle is tangent to the asymptotes of the hyperbola, so the distance |F1Q| is the radius of the circle.

|F1Q| = r = |Axo + byo + C|/square root of (A2 + B2)

Now we rewrite the asymptotes of the hyperbola, y = -(3/4) x, in general line equation, (3/4) x + y = 0. Compare the asymptotes with the general line equation, Ax + By + C = 0, we get, A = 3/4, B = 1 and C = 0.

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. r is the radius of the circle. Because the center of the circle is (-5, 0), so, we get xo = -5 and yo = 0.

r = |Axo + byo + C|/square root of (A2 + B2)
= |(3/4)(-5) + 1 × 0 + 0|/square root of ((3/4)2 + 12)
= |-15/4|/square root of (9/16 + 1)
= 15/4/square root of (9/16 + 16/16)
= 15/4/square root of 25/16
= 15/4 ÷ 5/4
= 15/4 × 4/5
= 3

So, the radius of the circle is 3. The circle equation is: (x + 5)2 + y2 = 32. Watch the video for more details.

how to find the circle equation when the center of the circle is the left focus of the hyperbola and the circle tangent to the asymptotes of the hyperbola?