Question

A pair of tangents are drawn from a point on the directrix to a parabola $y^2=4ax$. The angle formed by the tangents will always be ${90}^{\circ }$

• A. True
• B. False

Answer 1

The correct option is A

True

Let's draw a parabola first

p is the point on the directrix and A and B are the points where tangents from P touches the parabola.

Let $P\equiv (-a,y)$, both tangents at A point is,

The equation for tangent at A point is,

$ty=x+a{t}^2$

for tangent at A

$t_{1}y=-a+a{t}_1^2$

$y=\frac{a}{t_1}(t_1^2-1)$ ..............(1)

for tangent at B

$t_{2}y=-a+a{t}_2^2$

$y=\frac{a}{t_2}(t_2^2-1)$ ......(2)

Eliminating y

$t_1-\frac{1}{t_1}=t_2-\frac{1}{t_2}$

$t_1-t_2=\frac{1}{t_1}-\frac{1}{t_2}$

=$\frac{t_2-t_1}{t_1{t}_2}$

i.e., $t_1{t}_2=-1$

We know slope of a tangent at a point given by parameter is $\left(\frac{1}{t}\right)$

If 2 tangent are perpendicular then product of slope will be -1.

i.e., $\frac{1}{t_1}.\frac{1}{t_2}=-1$

$t_1{t}_2=-1$

$\therefore$ Tangent are at right angle to each other