A) \[\frac{1}{{{x}^{2}}}-\frac{9}{{{y}^{2}}}=8\]
B) \[\frac{1}{{{x}^{2}}}-\frac{8}{{{y}^{2}}}=9\]
C) \[\frac{8}{{{x}^{2}}}-\frac{1}{{{y}^{2}}}=9\]
D) \[\frac{2}{{{x}^{2}}}-\frac{3}{{{y}^{2}}}=8\]
Correct Answer: C
Solution :
[c] Let \[P=(h,k)\]. Then equation of chord of contact is \[\frac{hx}{2}-ky=1\] ...(1) Also, the equation of normal to hyperbola is \[\sqrt{2}x\cos \theta +y\cot \theta =3\] ...(2) Comparing (1) and (2), we get \[\tan \theta =-\frac{1}{3k},\,\,\sec \theta =\frac{2\sqrt{2}}{3h}\] Eliminating \[\theta ,\] we get \[\frac{8}{{{h}^{2}}}-\frac{1}{{{k}^{2}}}=9\] Therefore, required locus is \[\frac{8}{{{x}^{2}}}-\frac{1}{{{y}^{2}}}=9.\]
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