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  4. Square of the area of the triangle formed by end points of a focal chord P Q of length 32 units of the parabola y2=8 x and its vertex is

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Q. Square of the area of the triangle formed by end points of a focal chord $P Q$ of length $32$ units of the parabola $y^{2}=8 x$ and its vertex is

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Solution:

$P Q=2\left(t+\frac{1}{t}\right)^{2}=32$
image
$t+\frac{1}{t}=4,-4$
Area of $\Delta O P Q$ is $=\frac{1}{2}\begin{vmatrix}2 t^{2} & 4 t & 1 \\ \frac{2}{t^{2}} & -\frac{4}{t} & 1 \\ 0 & 0 & 1\end{vmatrix}$
$\left|\frac{1}{2}\left(-\frac{8}{t}-8 t\right)\right|=16$