A line touches a hyperbola at P and intersects the pair of asymptotes at points Q and R. If equation of pair of asymptotes is given by $4 x^{2} + 8 x y - y^{2} = 0$ and mid-point of QR is (1,0), then

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Equation of hyperbola is given by $4 x^{2} + 8 x y - y^{2} - 4 = 0$

If circumcenter of $Δ O Q R$ is (a, b) then a + 9b = 4, (O is center of hyperbola)

If circumcenter of $Δ O Q R$ is (a, b) then a – b = -1; (O is center of hyperbola)

Length of transverse axis of hyperbola is $\frac{\sqrt{89} - 3}{10}$

answer is A, C.

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Detailed Solution

(1,0) lies on hyperbola
Equation of tangent to hyperbola at (1,0) is $x + y - 1 = 0$
Homogenize the circle $x^{2} + y^{2} + 2 g x + 2 f y = 0$ with $x + y - 1 = 0$ and compare that with given pair of asymptotes

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