Find an equation of the hyperbola whose directrix is the normal to circle x 2 + y 2 – 4x – 6y + 9 = 0 having slope is 2 and eccentricity is equal to radius of given circle where focus of hyperbola is point of contact of given circle with y-axis.
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11x 2 – y 2 – 16xy –16x + 38y – 41 = 0
Sol. Let equation of directrix be y = 2 x + C ......(i)
(1) passes through centre (2, 3) of the circle
⇒ C = – 1
∴ (i) reduces to y = 2 x – 1
∴ equation of required hyperbola will be
= 2 
which reduces to ⇒ 11x 2 – y 2 – 16xy –16x + 38y – 41 = 0
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