A) \[{{({{x}^{2}}-{{y}^{2}})}^{2}}=6{{x}^{2}}+2{{y}^{2}}\]
B) \[{{({{x}^{2}}-{{y}^{2}})}^{2}}=6{{x}^{2}}-2{{y}^{2}}\]
C) \[{{({{x}^{2}}+{{y}^{2}})}^{2}}=6{{x}^{2}}+2{{y}^{2}}\]
D) \[{{({{x}^{2}}+{{y}^{2}})}^{2}}=6{{x}^{2}}-2{{y}^{2}}\]
Correct Answer: C
Solution :
Equation of ellipse, \[{{x}^{2}}+3{{y}^{2}}=6\] Equation of tangent of ellipse with slope m is |
\[y=mx\pm \sqrt{6{{m}^{2}}+2}\] |
\[m=-\frac{h}{k}\] |
and tangent passes through \[(h,k)\] |
\[\therefore \]\[k=h\left( -\frac{h}{k} \right)\pm \sqrt{\frac{6{{h}^{2}}}{{{k}^{2}}}+2}\]\[\Rightarrow \]\[{{k}^{2}}+{{h}^{2}}=\pm \sqrt{6{{h}^{2}}+2{{k}^{2}}}\]\[\Rightarrow \]\[{{({{h}^{2}}+{{k}^{2}})}^{2}}=6{{h}^{2}}+2{{k}^{2}}\]So, locus is \[({{x}^{2}}+{{y}^{2}})=6{{x}^{2}}+2{{y}^{2}}.\] |
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