A) \[{{\sin }^{-1}}(m)\]
B) \[{{\sin }^{-1}}\left( \frac{a}{bm} \right)\]
C) \[{{\sin }^{-1}}\left( \frac{b}{am} \right)\]
D) \[{{\sin }^{-1}}\left( \frac{bm}{a} \right)\]
Correct Answer: C
Solution :
| [c] Equation of tangent at point \['\varphi '\] on the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is |
| \[\frac{x}{a}\sec \varphi -\frac{y}{b}\tan \varphi =1\] or |
| \[y=\frac{b}{a}x\cos ec\,\varphi -b\cot \varphi \] ?. (1) |
| If \[y=mx+\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}\] ?. (2) |
| also touches the hyperbola then on comparing (1) & (2) |
| \[1=\frac{\frac{b}{a}\cos ec\,\varphi }{m}=\frac{-b\cot \,\,\varphi }{\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}}\] |
| Hence, \[m=\frac{b}{a}\cos ec\varphi ;\] or \[\cos ec\varphi =\frac{am}{b}\] |
| Or \[\sin \varphi =\frac{b}{am},\] or \[\varphi ={{\sin }^{-1}}\frac{b}{am}\] |
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