Bishesh Gautam

Let two tangents are drawn from point P.

Given equation of parabola is: y²=4ax.............(i)

We know that the locus of point P from which two perpendicular tangents are drawn to the parabola is the directrix of the parabola.

The standard equation of parabola (y - k)² = 4p(x-h) has focus (x+h, k) and the directrix is (h - p).

From (i), we get,

h=0, k=0, p=1

Directrix of (i) is x= 0 - 1 = -1

Hence, the required locus is x= -1

Here, the given equation of parabola is y²= 8x.

The equation of tangent to the parabola y²=8x is,   

               y= mx + 2/m

This tangent passes through the point (-2, 3)

So,       3 = -2m + 2/m

     or,  3m + 2m² = 2

    or,   2m² + 3m - 2 = 0

    or,   2m² + (4 - 1)m -2 = 0

    or,   2m² + 4m - m - 2 = 0

    or, ...