Question

The portion of a tangent to a parabola cut off between the directrix and the curve subtends right angle at the focus.

• A. True
• B. False

Answer 1

The correct option is A

True

Here we are required to find the angle betwwen PS and DS.

D is the point where the tangent at P meets the directrix.

We know that the equation of the tangent,

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

The tangent passes through a point D(-a,y') situated on the directrix

$t{y}^{\prime }=-a+a{t}^2$

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

$D\equiv (-a,\frac{a(t^2-1)}{t})$

Slope of SD=$\frac{a(t^2-1)}{t(-2a)}=-\frac{t^2-1}{2t}=m_1$

Slope of PS=$\frac{2at}{a(t^2-1)}=\frac{2t}{t^2-1}=m_2$

$\therefore m_1\times m_2=-1$

$\therefore$ The time segment PD extends right angle at the focus S.