Subtangent and Subnormal
Category : JEE Main & Advanced
Let the tangent and normal at \[P({{x}_{1}},{{y}_{1}})\]meet the x-axis at \[A\] and \[B\] respectively. Length of subtangent at \[P({{x}_{1}},{{y}_{1}})\]to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is
\[DA=CA-CD=\frac{{{a}^{2}}}{{{x}_{1}}}-{{x}_{1}}\].
Length of sub-normal at \[P({{x}_{1,}}{{y}_{1}})\]to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[BD=CD-CB={{x}_{1}}-\left( {{x}_{1}}-\frac{{{b}^{2}}}{{{a}^{2}}}{{x}_{1}} \right)\]\[=\frac{{{b}^{2}}}{{{a}^{2}}}{{x}_{1}}=(1-{{e}^{2}}){{x}_{1}}\].
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