equation of tangent y−√3x∣+√3=0 by equation of tangent Let slope =S=√3 Constant =−√3 By condition of tangency ⇒6=6a2−9a ⇒a=2,b2=9 Equation of Hyperbola is
x2
4
−
y2
9
=1 and for tangent Point of contact is (4,3√3)=(x0,y0) Now e =√1+
9
4
=
√13
2
Again product of focal distances m=(x0e+a)(x0e−a) m+4e2=20e2−a2 =20×
13
4
−4=61 (There is a printing mistake in the equation of directrix x=±
4
√3
. Corrected equation is x=±
4
√13
for directrix, as eccentricity must be greater than one, so question must be bonus)