Question

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

Answer 1

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 .