24) Find the equations of the tangent and normal to the hyperbola
\frac{x^2 }{a^2} - \frac{y^2 }{b^2 }= 1 at the point (x_0 , y_0 )

Answers (1)

Given equation is 
\frac{x^2 }{a^2} - \frac{y^2 }{b^2 }= 1 \Rightarrow y^2a^2 = x^2b^2 -a^2b^2
Now ,we know that
 slope of tangent = 2ya^2\frac{dy}{dx} = 2xb^2 \Rightarrow \frac{dy}{dx} = \frac{xb^2}{ya^2}
at point (x_0 , y_0 )
\frac{dy}{dx} = \frac{x_0b^2}{y_0a^2}
equation of tangent at point (x_0 , y_0 ) with slope \frac{xb^2}{ya^2}
y-y_1=m(x-x_1)\\ y-y_0=\frac{x_0b^2}{y_0a^2}(x-x_0)\\ yy_0a^2-y_0^2a^2 = xx_0b^2-x_0^2b^2\\ xx_0b^2 - yy_0a^2 = x_0^2b^2-y_0^2a^2
Now, divide  both sides by a^2b^2
\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = \left ( \frac{x_0^2}{a^2} - \frac{y_0^2}{b^2} \right )
                        =1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left ( \because \frac{x_0^2}{a^2}-\frac{y_0^2}{b^2 } = 1\right )
 \frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1
Hence, the equation of tangent is 

\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1                                                  
We know that
Slope \ of \ normal= \frac{-1}{slope \ of \ tangent } = -\frac{y_0a^2}{x_0b^2}
equation of normal  at the point (x_0 , y_0 ) with slope  -\frac{y_0a^2}{x_0b^2}
y-y_1=m(x-x_1)\\ y-y_0=-\frac{y_0a^2}{x_0b^2}(x-x_0)\\ \frac{y-y_0}{y_0a^2} + \frac{x-x_0}{x_0b^2} = 0

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