A) 4
B) 2
C) \[\frac{1}{2}\]
D) \[\frac{1}{4}\]
Correct Answer: A
Solution :
Equation of chord of hyperbola \[\frac{{{x}^{2}}}{2}\,-\frac{{{y}^{2}}}{1}=1,\]whose mid-point is (h, k) is \[\frac{hx}{2}-ky=\frac{{{h}^{2}}}{2}\,-\frac{{{k}^{2}}}{1}\] (using \[T={{S}_{1}}\] ) As it is tangent to the circle \[{{x}^{2}}+{{y}^{2}}=4,\] so \[\left| \frac{\frac{{{h}^{2}}}{2}-{{k}^{2}}}{\sqrt{\frac{{{h}^{2}}}{4}+{{k}^{2}}}} \right|\,=2\Rightarrow \,{{\left( \frac{{{h}^{2}}}{2-{{k}^{2}}} \right)}^{2}}\,=4\left( \frac{{{h}^{2}}}{4}\,+{{k}^{2}} \right)\] \[\Rightarrow \] Locus of (h, k) is \[{{({{x}^{2}}\,-2{{y}^{2}})}^{2}}=4({{x}^{2}}+4{{y}^{2}})\] \[\therefore \,\,\lambda =4\]You need to login to perform this action.
You will be redirected in
3 sec