Question

Show that the tangents at the ends of a focal chord of the parabola

y^{2} = 4 a x

intersect at right angle.

Accepted Answer

We need to prove that the tangents at the ends of a focal chord of the parabola $y^{2} = 4 a x$ intersects at right angle.

Let the coordinates of end points of the focal chord of parabola $y^{2} = 4 a x$ be $P (a t_{1}, 2 a {t_{1}}^{2})$ and $Q (a t_{2}, 2 a {t_{2}}^{2})$ , where $t_{1} \neq t_{2}$

And, the coordinates of focus is $S (a, 0)$ .

Since the points $P, S,$ and $Q$ are collinear.

$∴$ Slope of line $S P =$ Slope of line $S Q$

$\Rightarrow \frac{2 a t_{1} - 0}{a {t_{1}}^{2} - a} = \frac{2 a t_{2} - 0}{a {t_{2}}^{2} - a}$

$\Rightarrow \frac{2 t_{1}}{{t_{1}}^{2} - 1} = \frac{2 t_{2}}{{t_{2}}^{2} - 1}$

$\Rightarrow t_{1} ({t_{2}}^{2} - 1) = t_{2} ({t_{1}}^{2} - 1)$

$\Rightarrow t_{1} {t_{2}}^{2} - t_{1} = t_{2} {t_{1}}^{2} - t_{2}$

$\Rightarrow t_{1} {t_{2}}^{2} - t_{2} {t_{1}}^{2} - t_{1} + t_{2} = 0$

$\Rightarrow t_{1} t_{2} (t_{2} - t_{1}) + (t_{2} - t_{1}) = 0$

$\Rightarrow (t_{1} t_{2} + 1) (t_{2} - t_{1}) = 0$

$\Rightarrow t_{1} t_{2} = - 1 \to (i)$ (As, $t_{1} \neq t_{2}$ )

We know that the equation of tangent to the parabola $y^{2} = 4 a x$ at point $P (a {t_{1}}^{2}, 2 a t_{1})$ is given by

$t_{1} y = x + a {t_{1}}^{2}$

$\Rightarrow y = \frac{1}{t_{1}} x + a t_{1} \to (i i)$

Similarly, the equation of the tangent to the parabola at point $Q (a {t_{2}}^{2}, 2 a t_{2})$ is given by

$t_{2} y = x + a {t_{2}}^{2}$

$\Rightarrow y = \frac{1}{t_{2}} x + a t_{2} \to (i i i)$

Clearly, $(i i),$ and $(i i i)$ , are the equations of the tangents in the form, $y = m x + c$ , where $m$ is the slope of the line with intercept $' c'$ on the $y -$ axis.

Hence, the slope of the tangents $(i i),$ and $(i i i)$ are $\frac{1}{t_{1}}$ and $\frac{1}{t_{2}}$ respectively.

Now, $\frac{1}{t_{1}} \times \frac{1}{t_{2}} = \frac{1}{t_{1} t_{2}} = - 1$ [From $(i)$ ]

Since the product of slopes of tangents is $- 1$ .

Therefore, the tangents at the ends of a focal chord of the parabola $y^{2} = 4 a x$ intersects at right angle.

Show that the tangents at the ends of a focal chord of the parabola $y^{2} = 4 a x$ intersect at right angle.

Important Questions on Miscellaneous Exercise

Learn and Prepare for any exam you want !

Show that the tangents at the ends of a focal chord of the parabola y2=4ax intersect at right angle.

Important Questions on Miscellaneous Exercise

Learn and Prepare for any exam you want !

Show that the tangents at the ends of a focal chord of the parabola $y^{2} = 4 a x$ intersect at right angle.