The triangle PQR of area A is inscribed in the parabola $y^{2} = 4 a x$ such that P lies at the vertex of the parabola and base QR is a focal chord. The numerical difference of the ordinates of the points Q & R is

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Solution

The correct option is C
QR is a focal chord
$\Rightarrow (a t^{2}, 2 a t)$ $&$ $Q (\frac{a}{t^{2}}, - \frac{2 a}{t})$
$\Rightarrow d = ∣ ∣ 2 a t + \frac{2 a}{t} ∣ ∣ = 2 a ∣ ∣ t + \frac{1}{t} ∣ ∣$
$N o w A = \frac{1}{2}$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} a t^{2} & 2 a t & 1 \frac{a}{t^{2}} & - \frac{2 a}{t} & 1 0 & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = a^{2} ∣ ∣ t + \frac{1}{t} ∣ ∣$
$\Rightarrow 2 a ∣ ∣ t + \frac{1}{t} ∣ ∣ = \frac{2 A}{a}$

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The triangle PQR of area A is inscribed in the parabola y2=4ax such that P lies at the vertex of the parabola and base QR is a focal chord. The numerical difference of the ordinates of the points Q & R is

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