A) \[\left( \frac{2}{5},\ \frac{1}{5} \right)\]
B) \[\left( -\frac{2}{5},\ \frac{1}{5} \right)\]
C) \[\left( -\frac{2}{5},\ -\frac{1}{5} \right)\]
D) \[\left( \frac{2}{5},\ -\frac{1}{5} \right)\]
Correct Answer: B
Solution :
Ellipse is \[\frac{{{x}^{2}}}{\frac{1}{4}}+\frac{{{y}^{2}}}{\frac{1}{9}}=1\]Þ \[{{a}^{2}}=\frac{1}{4}\], \[{{b}^{2}}=\frac{1}{9}\] The equation of its tangent is \[4xx'+9yy'=1\] \\[m=-\frac{4x'}{9y'}=\frac{8}{9}\]Þ\[x'=-2y'\] and \[4x{{'}^{2}}+9y{{'}^{2}}=1\]Þ\[4x{{'}^{2}}+9\frac{x{{'}^{2}}}{4}=1\]Þ\[x'=\pm \frac{2}{5}\] When \[x'=\frac{2}{5},\,\,y'=-\frac{1}{5}\]and when \[x'=-\frac{2}{5},\,\,\,y'=\frac{1}{5}\] Hence points are \[\left( \frac{2}{5},\,\,-\frac{1}{5} \right)\,\]and \[\left( -\frac{2}{5},\,\,\frac{1}{5} \right)\].
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