Let P Q be a focal chord of the parabola y 2 = 36 x of length 100 , making an acute angle with the positive x - axis. Let the ordinate of P be positive and M be the point on the line segment P Q such that P M : M Q = 3 : 1 . Then which of the following points does NOT lie on the line passing…

Question

Let P Q be a focal chord of the parabola y 2 = 36 x of length 100 , making an acute angle with the positive x - axis. Let the ordinate of P be positive and M be the point on the line segment P Q such that P M : M Q = 3 : 1 . Then which of the following points does NOT lie on the line passing through M and perpendicular to the line P Q ?

Aicharya · Accepted Answer

We have been given the length of focal chord $P Q = 100$ .

Length of focal chord at $(t) = a {(t + \frac{1}{t})}^{2} = 100$
Where $a = 9$
$t + \frac{1}{t} = \pm \frac{10}{3}$
$∴ t = 3, \frac{1}{3}, - 3, \frac{- 1}{3}$
Since ordinate of $P$ is $+ ve$
$∴ t = 3$ or $\frac{1}{3}$
( $t = 3$ as slope of $P Q$ is positive)
$∴ M (21, 9)$
Required line $: y - 9 = \frac{- 80}{60} (x - 21)$
$\Rightarrow 4 x + 3 y = 111$
$(- 3, 43)$ does not lie on line.
Hence this is the correct option.

Let P Q be a focal chord of the parabola y 2 = 36 x of length 100 , making an acute angle with the positive…

Solution