Question

The triangle PQR of area ' $A$ ' is inscribed in the parabola $y ^2=4 ax$ such that the vertex $P$ lies at the vertex of the parabola and the base $QR$ is a focal chord. The modulus of the difference of the ordinates of the points $Q$ and $R$ is -

• A. $\frac{ A }{2 a }$
• B. $\frac{ A }{ a }$
• C. $\frac{2 A }{ a }$
• D. $\frac{4 A }{ a }$

Answer 1

Answer: (C)

Since $Q R$ is focal chord so vertex of $Q$ is (at ${ }_1^2, 2 at _1$ ) and $R$ is $\left( at _2^2, 2 at _2\right)$
area of $\Delta PQR =\frac{1}{2}\begin{vmatrix}0 & 0 & 1 \\ at _1^2 & 2 at _1 & 1 \\ at _2^2 & 2 at _2 & 1\end{vmatrix}$
$A=\frac{1}{2}\left|2 a^2 t_1^2 t_2-2 a^2 t_1 t_2^2\right|$
$A=\frac{a}{2}\left|2 at _1-2 a t_2\right| \left[t_1 t_2=-1\right]$