The circle x 2 + y 2 – 8x = 0 and hyperbola 
– 
= 1 intersect at the points A and B.
(i) Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
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(i) Let equation of tangent to hyperbola
2sec θ x – 3 tan θ y = 6
It is also tangent to circle x 2 + y 2 – 8x = 0
⇒ 
= 4
(8sec θ – 6) 2 = 16 (13sec 2 θ – 9)
⇒ 12sec 2 θ + 8sec θ – 15 = 0
⇒ sec θ = 
and 
but sec ≠ 
⇒ sec θ = 
⇒ tan θ = – 
so that slope is positive
Equation of tangent = 2x – 
y + 4 = 0
(i) x 2 + y 2 – 8x = 0
– 
= 1 ⇒ 4x 2 – 9y 2 = 36
⇒ 4x 2 – 9(8x – x 2 ) = 36
13x 2 – 72x – 36 = 0
13x 2 – 78x + 6x – 36 = 0
(13x + 6) (x – 6) = 0
⇒ x = – 
and x = 6
But x > 0 ⇒ x = 6
⇒ A(6, 
) and B (6, – 
)
⇒ Equation of circle with AB as a diameter x 2 + y 2 – 12x + 24 = 0
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