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

Question

Let $PQ$ be a focal chord of the parabola ${y}^{2}=36x$ 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 PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $PQ$?

Accepted Answer

The given parabola is of the form ${y}^{2}=4ax$. Here, 4a = 36, which means a = 9.

Length of focal chord at $(t)=a{(t+\frac{1}{t})}^{2}=100$

Where $a=9$

$\begin{gathered} t+\frac{1}{t}=\pm \frac{10}{3} \ 	herefore \, t=3,\frac{1}{3},-3,\frac{-1}{3} \end{gathered}$

Since ordinate of $P$ is $+ve$

$	herefore t=3\frac{1}{3}$

For each value of t, we can find the coordinates of the points P and Q on the parabola using the parametric form of the parabola ${y}^{2}=4ax$, where a is 9. The parametric form is given by $x=a{t}^{2}$ and $y=2at$.

Let's consider $t=3$ first.

For $t=3$, the coordinates of P are given by $P(a{t}^{2},2at)=P(81,54)$.

Since ${t}_{1}{t}_{2}=-1$ for a focal chord, the parameter value for Q is ${t}_{2}=-1/{t}_{1}=-1/3$. So the coordinates of Q are given by $Q(a{t}_{2}^{2},2a{t}_{2})=Q(1,-6)$.

Now, let's find the coordinates of the point M that divides the line segment PQ in the ratio 3:1.

The x-coordinate of M is given by $\frac{3{Q}_{x}+{P}_{x}}{4}=\frac{3\ast 1+81}{4}=21$.

The y-coordinate of M is given by $\frac{3{Q}_{y}+{P}_{y}}{4}=\frac{3\ast (-6)+54}{4}=9$.

So, the coordinates of M are M$(21,9)$.

We can repeat these steps for $t=1/3$ to find the other set of points P, Q, and M. However, since the ordinate of P is positive and PQ makes an acute angle with the positive x-axis, $t=3$ is the appropriate choice in this context.

The slope of the line PQ is ${m}_{PQ}=\frac{-6-54}{1-81}=\frac{-60}{-80}=\frac{3}{4}$. The slope of the line perpendicular to PQ is ${m}_{\perp PQ}=-1/{m}_{PQ}=-\frac{4}{3}$. Thus, the equation of the line through M and perpendicular to PQ is $(y-9)=-\frac{4}{3}(x-21)$, or $4x+3y=111$.

Now, we check which of the given points do NOT lie on this line.

Option A : $(6,29)$
Substitute these values into the equation :

$4	imes 6+3	imes 29=111?$

$24+87=111?$

$111=111$

So, option A does lie on the line.

Option B : $(-3,43)$
Substitute these values into the equation :

$4	imes (-3)+3	imes 43=111?$

$-12+129=111?$

$117=111$

So, option B does NOT lie on the line.

Option C : $(3,33)$
Substitute these values into the equation :

$4	imes 3+3	imes 33=111?$

$12+99=111?$

$111=111$

So, option C does lie on the line.

Option D : $(-6,45)$
Substitute these values into the equation :

$4	imes (-6)+3	imes 45=111?$

$-24+135=111?$

$111=111$

So, option D does lie on the line.

Therefore, $(-3,43)$ which does not lie on the line passing through M and perpendicular to the line PQ.

Solutions · 1