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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Hyperbola

NTA Abhyas 2020 · Mathematics · Hyperbola

From a point P, tangents are drawn to the curve x ^ 2 2 - y ^ 2 = 1...

From a point P,P, tangents are drawn to the curve x22−y2=1.\dfrac{{x}^{2}}{2} - {y}^{2} = 1 . If the chord of contact is a normal chord and the locus of PP is the curve 8x2−1y2=10λ,\dfrac{8}{{x}^{2}} - \dfrac{1}{{y}^{2}} = 10\lambda, then the value of λ\lambda is equal to

Correct answer

0.9

Step-by-step solution

Let P(h,k)P\left(h, k\right), then equation of chord of contact is hx2−ky=1……(i)\dfrac{hx}{2} - ky = 1 \ldots \ldots \left(i\right)
And equation of normal to hyperbola is 2xcosϕ+ycotϕ=3……(ii)\sqrt{2}xcos ϕ + ycot ϕ = 3 \ldots \ldots \left(ii\right)
On comparing \left(i\right)and(ii)and \left(ii\right) , we get,
cotϕ=−3k,secϕ=223hcot ϕ = - 3k, sec ϕ = \dfrac{2\sqrt{2}}{3h}
Now eliminating ϕ,ϕ, we get,
8h2−1k2=9∴λ=0.9\dfrac{8}{{h}^{2}} - \dfrac{1}{{k}^{2}} = 9 ∴ \lambda = 0.9
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