A) \[\frac{11}{3}\]
B) \[-\frac{11}{3}\]
C) \[\frac{13}{2}\]
D) \[-\frac{13}{2}\]
Correct Answer: D
Solution :
let the coordinate at point of intersection of normal at P and Q be \[\left( h,k \right)\] |
Since, equation of normal to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at point \[\left( {{x}_{1}},{{y}_{1}} \right)\] is \[\frac{{{a}^{2}}x}{{{x}_{1}}}+\frac{{{b}^{2}}y}{{{y}_{1}}}\]\[={{a}^{2}}+{{b}^{2}}\] therefore equation of normal to the hyperbola \[\frac{{{x}^{2}}}{{{3}^{2}}}-\frac{{{y}^{2}}}{{{2}^{2}}}=1\] at point \[\operatorname{P}\left( 3\sec \operatorname{q},2tan\theta \right)\] is |
\[\frac{{{3}^{2}}x}{3\sec \theta }+\frac{{{2}^{2}}y}{2\tan \theta }={{3}^{2}}+{{2}^{2}}\]\[\Rightarrow 3x\cos \theta +2y\cot \theta ={{3}^{2}}+{{2}^{2}}..(1)\] |
Similarly, equation of normal to the hyperbola \[\frac{{{x}^{2}}}{{{3}^{2}}}-\frac{{{y}^{2}}}{{{2}^{2}}}\]at point \[\operatorname{Q}\left( 3\sec \phi ,2tan\phi \right)\]is \[\frac{{{3}^{2}}x}{3\sec \phi }+\frac{{{2}^{2}}y}{2\tan \phi }={{3}^{2}}+{{2}^{2}}\]\[\Rightarrow 3x\cos \phi +2y\cot \phi ={{3}^{2}}+{{2}^{2}}..(2)\] |
Given \[\theta +\phi =\frac{\pi }{2}\Rightarrow \phi =\frac{\pi }{2}-\theta \] and these passes through \[\left( h,k \right)\] |
\[\therefore \]from eq. (2) \[3x\cos \left( \frac{\pi }{2}-\theta \right)+2y\cot \left( \frac{\pi }{2}-\theta \right)={{3}^{2}}+{{2}^{2}}\]\[\Rightarrow 3h\cos \theta +2k\cot \theta ={{3}^{2}}+{{2}^{2}}..(3)\] |
and \[3h\cos \theta +2k\cot \theta ={{3}^{2}}+{{2}^{2}}\,\,\,\,\,\,\,\,..(4)\] |
Comparing equation (3) & (4), we get \[3h\cos \theta +2k\cot \theta =3h\sin \theta +2k\tan \theta \] |
\[3h\cos \theta -3h\sin \theta =2k\tan \theta -2k\cot \theta \] |
\[3h\left( \cos \theta -\sin \theta \right)=2k\left( \tan \theta -\cot \theta \right)\] |
\[3h\left( \cos \theta -\sin \theta \right)\]\[=2k\frac{\left( \sin \theta -\cos \theta \right)\left( \sin \theta +\cos \theta \right)}{\sin \theta \cos \theta }\] |
\[\operatorname{or}\,\,\,3h=\frac{-2k\left( \sin \theta +\cos \theta \right)}{\sin \theta \cos \theta }..(5)\] |
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