Let H : x 2 / a 2 − y 2 / b 2 = 1 have the same eccentricity as its conjugate hyperbola. The foot from a focus of H to a tangent is ( 1/2 , 15 /2 ) . A circle C : x 2 + y 2 = a 2 − 4 b 2 + 12 is als

Question

Let H : x 2 / a 2 − y 2 / b 2 = 1 have the same eccentricity as its conjugate hyperbola. The foot from a focus of H to a tangent is ( 1/2 , 15 ​ /2 ) . A circle C : x 2 + y 2 = a 2 − 4 b 2 + 12 is also given. Find the number of common tangents of H and C .

TestHub · Accepted Answer

Equal eccentricities give a = b . The tangent-foot lies on the auxiliary circle, so a 2 = ( 1/2 ) 2 + ( 15 ​ /2 ) 2 = 4 and a = b = 2 , giving C : x 2 + y 2 = 4 and two common tangents. Following the source construction, the answer is 2.

Mathematics - Hyperbola Question with Solution | TestHub

Options:

Doubts & Discussion