Name: Conic Sections: The Parabola
Uploaded: 2019-07-19
Duration: 9 min 43 s
Description: This video defines a parabola and explains how to graph a parabola in standard form, which will help you to solve questions in exam. Watch and learn.

Question

Find the condition that the chord

t_{1} t_{2}

of the parabola

y^{2} = 4 a x

passes through the point

(a, 3 a)

. Find the locus of intersection of the tangents at

t_{1}

and

t_{2}

under this condition.

Accepted Answer

Let the two points $t_{1}$ and $t_{2}$ are $A (a t_{1}, 2 a {t_{1}}^{2})$ and $B (a t_{2}, 2 a {t_{2}}^{2})$ of the parabola $y^{2} = 4 a x$ .

So, equation of the chord of parabola $y^{2} = 4 a x$ is $(y - 2 a t_{1}) = \frac{2 a t_{2} - 2 a t_{1}}{a t_{2} - a t_{1}} (x - a t_{1})$ ,

In the question,given that this chord passes through the point $(a, 3 a)$ ,

again, $(3 a - 2 a t_{1}) = \frac{2 a t_{2} - 2 a t_{1}}{a {t^{2}}_{2} - a {t^{2}}_{1}} (a - a {t^{2}}_{1})$

$\Rightarrow (3 a - 2 a t_{1}) = \frac{2 a (t_{2} - t_{1})}{a ({t_{2}}^{2} - {t_{1}}^{2})} (a - a {t_{1}}^{2})$

$\Rightarrow (3 a - 2 a t_{1}) = \frac{2}{t_{1} + t_{2}} (2 a - 2 a {t_{1}}^{2})$

$\Rightarrow (3 a - 2 a t_{1}) (t_{1} + t_{2}) = (2 a - 2 a {t_{1}}^{2})$

on simplification above equation, we get $2 - 3 (t_{1} + t_{2}) + 2 t_{1} t_{2} = 0$

This is the required condition of chord.

Further, Equation of tangent to the parabola $y^{2} = 4 a x$ at $t$ is given by , $t y = x + a t^{2}$

Thus equation of tangent to the parabola at $t_{1}$ and $t_{2}$ are given by $t_{1} y = x + a {t_{1}}^{2}$ ...... $(1)$ and $t_{2} y = x + a {t_{2}}^{2}$ ........ $(2)$

Subtract $(2)$ from $(1)$

$t_{1} y - t_{2} y = a {t_{1}}^{2} - a {t_{2}}^{2}$

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

Substitute value of y in $(1)$

$\Rightarrow a t_{1} (t_{1} + t_{2}) = x + a {t_{1}}^{2}$

$\Rightarrow x = a t_{1} t_{2}$

Hence, the point of intersection is $(a t_{1} t_{2}, a (t_{1} + t_{2}))$

Find the condition that the chord $t_{1} t_{2}$ of the parabola $y^{2} = 4 a x$ passes through the point $(a, 3 a)$ . Find the locus of intersection of the tangents at $t_{1}$ and $t_{2}$ under this condition.

Important Questions on Parabola

Learn and Prepare for any exam you want !

Find the condition that the chord t1t2 of the parabola y2=4ax passes through the point a,3a. Find the locus of intersection of the tangents at t1 and t2 under this condition.

Important Questions on Parabola

Learn and Prepare for any exam you want !

Find the condition that the chord $t_{1} t_{2}$ of the parabola $y^{2} = 4 a x$ passes through the point $(a, 3 a)$ . Find the locus of intersection of the tangents at $t_{1}$ and $t_{2}$ under this condition.