11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer If the normal at any point P on the ellipse cuts the major and minor axes in G and g  respectively and C be the centre of the ellipse, then [Kurukshetra CEE 1998]

    A)            \[{{a}^{2}}{{(CG)}^{2}}+{{b}^{2}}{{(Cg)}^{2}}={{({{a}^{2}}-{{b}^{2}})}^{2}}\]

    B)            \[{{a}^{2}}{{(CG)}^{2}}-{{b}^{2}}{{(Cg)}^{2}}={{({{a}^{2}}-{{b}^{2}})}^{2}}\]

    C)            \[{{a}^{2}}{{(CG)}^{2}}-{{b}^{2}}{{(Cg)}^{2}}={{({{a}^{2}}+{{b}^{2}})}^{2}}\]

    D)            None of these

    Correct Answer: A

    Solution :

               Let at a point \[({{x}_{1}},{{y}_{1}})\] normal will be                   \[\frac{(x-{{x}_{1}}){{a}^{2}}}{{{x}_{1}}}=\frac{(y-{{y}_{1}}){{b}^{2}}}{{{y}_{1}}}\]                   At \[G,\,\,y=0\]Þ\[x=CG=\frac{{{x}_{1}}({{a}^{2}}-{{b}^{2}})}{{{a}^{2}}}\]                   At \[g,\,\,\,x=0\]Þ\[y=Cg=\frac{{{y}_{1}}({{b}^{2}}-{{a}^{2}})}{{{b}^{2}}}\]                    \[\frac{x_{1}^{2}}{{{a}^{2}}}+\frac{y_{1}^{2}}{{{b}^{2}}}=1\] Þ\[{{a}^{2}}{{(CG)}^{2}}+{{b}^{2}}{{(Cg)}^{2}}={{({{a}^{2}}-{{b}^{2}})}^{2}}.\]


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