JEE Main 2023 — Parabola Question with Solution

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

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

Where $a=9$

$$\begin{array}{c}t+\frac{1}{t}=\pm \frac{10}{3}\\ \\ \therefore t=3,\frac{1}{3},-3,\frac{-1}{3}\end{array}$$

Since ordinate of $P$ is $+\mathrm{ve}$

$$\therefore t=3\text{ or }\frac{1}{3}$$

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.

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

2. 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.

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

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

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 :

$4\times 6+3\times 29=111?$

$24+87=111?$

$111=111$

So, option A does lie on the line.

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

$4\times (-3)+3\times 43=111?$

$-12+129=111?$

$117=111$

So, option B does NOT lie on the line.

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

$4\times 3+3\times 33=111?$

$12+99=111?$

$111=111$

So, option C does lie on the line.

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

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

$-24+135=111?$

$111=111$

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.