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  • Let <img alt="\mathrm{\mathrm{P}(\mathrm{a} \sec \theta, \mathrm{b} \tan \theta)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cmathrm%7BP%7D%28%5Cmathrm%7Ba%7D%20%5Csec

Let \mathrm{\mathrm{P}(\mathrm{a} \sec \theta, \mathrm{b} \tan \theta)} and \mathrm{Q(a \sec \phi, b \tan \phi)} ,where \mathrm{\text {, where } \theta+\phi=\frac{\pi}{2}},  be two points on the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}  .

 If (h, k) is the point of the intersection of the normals at P and Q, then k is equal to

 

Option: 1

\mathrm{\frac{a^2+b^2}{a}}


Option: 2

\mathrm{-\left(\frac{a^2+b^2}{a}\right)}


Option: 3

\mathrm{\frac{a^2+b^2}{b}}


Option: 4

\mathrm{-\left(\frac{\mathrm{a}^2+\mathrm{b}^2}{\mathrm{~b}}\right)}


Answers (1)

best_answer

Equation of normal at \mathrm{P(\theta)} is 

\mathrm{ a x+b y \operatorname{cosec} \theta=\left(a^2+b^2\right) \sec \theta}-----------(1)

And \mathrm{Q(\phi)}is

\mathrm{\begin{aligned} & a x+b y \operatorname{cosec} \phi=\left(a^2+b^2\right) \sec \phi \\ & \Rightarrow a x+b y \sec \theta=\left(a^2+b^2\right) \operatorname{cosec} \theta \end{aligned} }----------(2)

From (1) and (2)   \mathrm{y=-\frac{a^2+b^2}{b}=k}

Posted by

Shailly goel

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