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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.