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# Solution for Find the Equation of the Tangent and the Normal to the Following Curve at the Indicated Point X 2 a 2 + Y 2 B 2 = 1 at ( X 1 , Y 1 ) ? - CBSE (Science) Class 12 - Mathematics

#### Question

Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)$ ?

#### Solution

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

$\text { Differentiating both sides w.r.t.x },$

$\frac{2x}{a^2} + \frac{2y}{b^2}\frac{dy}{dx} = 0$

$\Rightarrow \frac{2y}{b^2}\frac{dy}{dx} = \frac{- 2x}{a^2}$

$\Rightarrow \frac{dy}{dx} = \frac{- x b^2}{y a^2}$

$\text { Slope of tangent,}m= \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{- x_1 b^2}{y_1 a^2}$

$\text { Equation of tangent is },$

$y - y_1 = m \left( x - x_1 \right)$

$\Rightarrow y - y_1 = \frac{- x_1 b^2}{y_1 a^2}\left( x - x_1 \right)$

$\Rightarrow y y_1 a^2 - {y_1}^2 a^2 = - x x_1 b^2 + {x_1}^2 b^2$

$\Rightarrow x x_1 b^2 + y y_1 a^2 = {x_1}^2 b^2 + {y_1}^2 a^2 . . . \left( 1 \right)$

$\text { Since }\left( x_1 , y_1 \right)\text { lies on the given curve.Therefore},$

$\frac{{x_1}^2}{a^2} + \frac{{y_1}^2}{b^2} = 1$

$\Rightarrow \frac{{x_1}^2 b^2 + {y_1}^2 a^2}{a^2 b^2} = 1$

$\Rightarrow {x_1}^2 b^2 + {y_1}^2 a^2 = a^2 b^2$

$\text { Substituting this in (1), we get }$

$x x_1 b^2 + y y_1 a^2 = a^2 b^2$

${\text { Dividing this by } a}^2 b^2 ,$

$\frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1$

$\text { Equation of normal is },$

$y - y_1 = m \left( x - x_1 \right)$

$\Rightarrow y - y_1 = \frac{y_1 a^2}{x_1 b^2}\left( x - x_1 \right)$

$\Rightarrow y x_1 b^2 - x_1 y_1 b^2 = x y_1 a^2 - x_1 y_1 a^2$

$\Rightarrow x y_1 a^2 - y x_1 b^2 = x_1 y_1 a^2 - x_1 y_1 b^2$

$\Rightarrow x y_1 a^2 - y x_1 b^2 = x_1 y_1 \left( a^2 - b^2 \right)$

$\text { Dividing by } x_1 y_1$

$\frac{a^2 x}{x_1} - \frac{b^2 y}{y_1} = a^2 - b^2$

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Solution for question: Find the Equation of the Tangent and the Normal to the Following Curve at the Indicated Point X 2 a 2 + Y 2 B 2 = 1 at ( X 1 , Y 1 ) ? concept: Tangents and Normals. For the courses CBSE (Science), CBSE (Commerce), CBSE (Arts), PUC Karnataka Science
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