The portion of a tangent to a parabola cut off between the directrix and point of contact on the curve subtends _____ degree angle at the focus.

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Solution

Let's consider a standard parabola $y^{2} = 4 a x$ and a point $P (a t^{2}, 2 a t)$

Equation of tangent

T=0

$y . (2 a t) = 2 a (x + a t^{2})$

$t y = x + a t^{2}$ ...............(1)

Equation of directrix

$x = - a$ ...............(2)

Intersection of tangent $&$ directrix

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

$y = \frac{a t^{2} - a}{t}$

Coordinates of point $A (- a, \frac{a t^{2} - a}{t})$

Now, we have coordinates of A,P $&$ S.

We can find the slope of line AS $&$ PS, and then find the angle ASP.

Slope of line $A S (m_{1}) = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{\frac{a t^{2} - a}{t} - 0}{- 9 - a} = \frac{(\frac{t^{2} - 1}{t})}{- 2}$

$m 1 = \frac{1 - t^{2}}{2 t}$

Slope of line $P S (m_{2}) = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{2 a t - 0}{a t^{2} - a} = \frac{2 t}{t^{2} - 1}$

We see that

$m_{1} m_{2} = - 1$

$= \frac{1 - t^{2}}{2 t} \times \frac{2 t}{t^{2} - 1} = - 1$

So, we can say that $∠ A S P = 90^{\circ}$ OR the portion of atangent

to a parabola cut off between the directrix and point of

contact on the curve subtends a right angle at focus.

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Similar questions

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

y^{2} = 4 a x

θ

θ =

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

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