Q.298. To find equation of hyperbola given the equations of its asymptotes and a tangent.

Question 298.
Find the equation of a hyperbola with asymptotes x - 1 = 0 and 2x - y + 1 = 0
and a tangent line 4x + y + 5 = 0.

Answer 298.
The eqn. of hyperbola is
(x-1)(2x-y+1) = k
If 4x + y + 5 = 0 is the tangent, solution of the two equations should be unique
=> (x - 1)(2x+4x+5+1) = k should have one value of x
=> 6x^2 - 6 = k
=> k = - 6
=> eqn. of hyperbola is
(x - 1)(2x - y + 1) = - 6
=> 2x^2 - xy - x + y + 5 = 0

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