Question

the equation of a tangent to the parabola y2 = 8x is y = x + 2. the point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

Answer 1

y² = 8x

The equation of the tangent formed would be = y = x + 2

And the slope of the tangent is m = 1

and here when the 2 lines are perpendicular to each other the product of their slopes would be -1

m.m₁ = -1

m₁ = -1/1

= -1

now we will let the tangent be

y = m₁ x + c

= -x + c

A line which touches the parabola is said to be a tangent to the parabola

If we have a line y = m₁ x + c which will touch a parabola y² = 4 ax

we know c = a/m₁

y² = 8x and y = -x + c

4a = 8

a = 2

c = a/m₁ = 2/-1 = -2

the other tangent would be y = -x-2

The point of interest for both the tangents would be

y = x +2 -- (1)

y = -x -2 -- (2)

From the 1 and 2 we will get the following.

x + 2 = -x -2

2x = -4

x = -2

y = x + 2

= -2 +2 = 0

so the point would be (-2,0)