11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer Let P be a variable point on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] with foci \[{{F}_{1}}\] and \[{{F}_{2}}\]. If A is the area of the triangle \[P{{F}_{1}}{{F}_{2}}\], then maximum value of A is [IIT 1994; Kerala (Engg.) 2005]

    A)            ab   

    B)            abe

    C)            \[\frac{e}{ab}\]                         

    D)            \[\frac{ab}{e}\]

    Correct Answer: B

    Solution :

               \[b\sqrt{{{a}^{2}}-{{b}^{2}}}\]if \[a>b;a\sqrt{{{b}^{2}}-{{a}^{2}}}\]if \[P\] Area of \[P{{F}_{1}}{{F}_{2}}=\frac{1}{2}({{F}_{1}}{{F}_{2}})\times PL\]                     \[=\frac{1}{2}(2ac)\times y=ae.\frac{b}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}\]                   \[A=eb\sqrt{{{a}^{2}}-{{x}^{2}}}\], which is maximum when\[x=0\].                    Thus the maximum value of A is abe.

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