The locus of the poles of focal chords of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the focal chord pass through the focus $S(a, 0)$ and connect points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ on the parabola $y^2 = 4ax$.The

Vedclass · Accepted Answer

the directrix

Vedclass · Answer

(C) Let the focal chord pass through the focus $S(a, 0)$ and connect points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ on the parabola $y^2 = 4ax$.The equation of the chord $PQ$ is $y(t_1 + t_2) = 2x + 2at_1t_2$.Since it passes through $(a, 0)$,we have $0 = 2a + 2at_1t_2$,which implies $t_1t_2 = -1$.Let $(x_1, y_1)$ be the pole of this chord with respect to the parabola $y^2 = 4ax$.The equation of the polar of $(x_1, y_1)$ is $yy_1 = 2a(x + x_1)$,or $yy_1 - 2ax = 2ax_1$.Comparing this with the chord equation $y(t_1 + t_2) - 2x = 2at_1t_2$,we get:$\frac{t_1 + t_2}{y_1} = \frac{-2}{-2a} = \frac{2at_1t_2}{2ax_1}$From $\frac{-2}{-2a} = \frac{2at_1t_2}{2ax_1}$,we get $x_1 = at_1t_2$.Since $t_1t_2 = -1$,we have $x_1 = -a$.Thus,the locus of the pole is $x = -a$,which is the directrix of the parabola.

The locus of the poles of focal chords of the parabola $y^2 = 4ax$ is

Similar Questions

Let $f$ be a non-zero real-valued continuous function satisfying $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$. If $f(2)=9$,then $f(6)$ is equal to

Which one of the following is an artificial sweetening agent?

What is formed when carbon monoxide reacts with $Cl_2$ in the presence of sunlight?

The coils of a step-down transformer have $500$ and $5000$ turns. In the primary coil,an $AC$ of $4 \, A$ at $2200 \, V$ is sent. The value of the current and potential difference in the secondary coil will be:

Yeast is used in the production of

Vedclass Test Series

Exam Paper Generator

Online Exam Module