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Question

# Find the equations of the tangent and normal to the parabola y 2 = 4 ax at the point ( at 2 , 2 at ).

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Solution

## The equation of the given parabola is, y 2 =4ax Equation of the slope of the line is given by, slope= dy dx Hence, slope of the parabola is determined by differentiating with respect to x. 2y( dy dx )=4a dy dx = 2a y Slope of the tangent at point ( a t 2 ,2at ) is given by, ( dy dx ) ( a t 2 ,2at ) = 2a 2at = 1 t Thus, the equation of the tangent at ( a t 2 ,2at ) with slope 1 t is, y−at= 1 t ( x−a t 2 ) ty=x+a t 2 Slope of the normal at ( a t 2 ,2at ) is given by, −1 slope of tangent at ( a t 2 ,2at ) =−t Hence, the equation of normal at ( a t 2 ,2at ) with slope −t is given by, y−2at=−t( x−a t 2 ) y=−tx+2at+a t 2 Thus, the equation of the tangent is ty=x+a t 2 and equation of the normal is y=−tx+2at+a t 2 .

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