A variable tangent to the hyperbola 
meets the transverse axis and the tangent at the vertex (a, 0) at Q and R. If the locus of mid point of QR is
λ x (ky 2 + mb 2 ) = ab 2 then value of k 2 – ( λ + m) 2 is equal to-
Text Solution
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Equation tangent at the point θ is 
sec θ – 
tan θ = 1. It meets the transverse axis y = 0 and the tangent at vertex x = a in the point Q (a cos θ , 0) and R 
. If (h, k) is mid point of QR then 2h = a + a cos θ
⇒ cos θ = 
and 2k = 
and 4hk = 
= ab sin θ
⇒ 16h 2 k 2 = a 2 b 2 (1 – cos 2 θ )
= a 2 b 2 
⇒ 4hk 2 + b 2 h = b 2 a locus x(4y 2 + b 2 ) = ab 2 ⇒ λ = 1, k = 4, m = 1
∴ k
2 – ( λ + m) 2 = (4) 2 – (1 + 1) 2 = 16 – 4 = 12.
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