If the line y = m x + c is a common tangent to the hyperbola x 2 100 - y 2 64 = 1 and the circle x 2 + y 2 = 36 , then which one of the following is true?

Step-by-step solution

A line y = m x + c is a tangent to the hyperbola x 2 a 2 - y 2 b 2 = 1 if c 2 = a 2 m 2 - b 2 and the line is tangent to the circle x 2 + y 2 = r 2 is c 2 = r 2 1 + m 2 . Hence, for the given line and the hyperbola we have, c 2 = 100 m 2 − 64     . . . i and for the line and the circle, we have c 2 = 36 1 + m 2     . . . i i On comparing the above two equations, we get 100 m 2 - 64 = 36 + 36 m 2 64 m 2 = 100 m 2 = 100 64 ⇒   m = 10 8 = 5 4 . ⇒   c 2 = 36 1 + 25 16 = 3