Tangent circle to a hyperbola
Suppose we have the hyperbola . We want to place a circle of radius , and center at such that it passes through the point , while being tangent to the hyperbola at , where . Enter the value of
The answer is 14.
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1 solution
The circle has equation ( x − c ) 2 + y 2 = R 2 If we substitute y 2 = x 2 + 1 from the hyperbola, we obtain the quadratic in x :
( x 2 − 2 c x + c 2 ) + ( x 2 + 1 ) = R 2 ⇒ 2 x 2 − 2 c x + ( 1 + c 2 − R 2 ) = 0 ⇒ x = 4 2 c ± 4 c 2 − 4 ( 2 ) ( 1 + c 2 − R 2 ) = 2 c ± 2 R 2 − c 2 − 2 (i).
If these curves intersect in one value of x , then we require 2 R 2 − c 2 − 2 = 0 (ii). If the circle contains the point ( 1 , 0 ) , then we have ( 1 − c ) 2 = R 2 (iii). This system of equations ultimately gives us c = 4 , R = 3 , which turn produce x = 2 , y 2 = 5 . Hence, c + R + x + y 2 = 4 + 3 + 2 + 5 = 1 4 .