11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-3

  • question_answer
    The length of major axis of ellipse is 26 and its foci is \[\left( \pm \text{ }\mathbf{5},\mathbf{0} \right)\] then equation of ellipse be

    A)  \[\frac{{{x}^{2}}}{144}+\frac{{{y}^{2}}}{169}=1\]

    B)         \[\frac{{{x}^{2}}}{169}-\frac{{{y}^{2}}}{144}=-1\]

    C)  \[\frac{{{x}^{2}}}{169}+\frac{{{y}^{2}}}{144}=-1\]          

    D)  \[-\frac{{{x}^{2}}}{169}-\frac{{{y}^{2}}}{144}=-1\]

    Correct Answer: C

    Solution :

    [c] \[\because 2a=26~~~a\Rightarrow 13.\] foci\[=\left( \pm 5,0 \right)\] \[\therefore a.e=5\Rightarrow e=\frac{5}{a}=\frac{5}{13}\] \[\therefore {{b}^{2}}={{a}^{2}}(1-{{e}^{2}})=169{{\left[ 1-\left( \frac{25}{13} \right) \right]}^{2}}\] \[=169\times \frac{144}{169}=144.\] hence the equation of ellipse be  \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\Rightarrow \frac{{{x}^{2}}}{169}+\frac{{{y}^{2}}}{144}=1\] Hence, Option [b] is correct

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