Question

The equation of the tangent to the parabola
$y^2=8x\text{is}y=x+2$ . The point on this line from which the other tangent to the parabola is perpendicular to the given tangent.

Answer 1

Slope of the given tangent $=1$

$\Rightarrow$ slope the perpendicular tangent

$=\frac{-1}{1}=-1$ using $(m_1{m}_2=-1)$ for perpendicular lines

tangent of a parabola

$y=mn+a/m$

$m=-1$

$a=2$ comparing $y^2=8x$

with $y^2=4ax$

$\Rightarrow y=-x,-2$

and $y=x+2$ & $y=-x-2$ intersect at

their solution

$y=x+2$ (1)

$y=-x-2$ (2)

$\left(1\right)+\left(2\right)\Rightarrow 2y=0$

$=y=0$

substituting $y=0$ in $\left(1\right)\Rightarrow x=-2$

$\Rightarrow At(-2,0)$