Suppose AB is a focal chord of the parabola of length and slope . If the distance of the chord AB from the origin is d, then is equal to __________. [2024]

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Solution

Q.
Suppose AB is a focal chord of the parabola $y^{2} = 12 x$ of length $l$ and slope $m < \sqrt{3}$ . If the distance of the chord AB from the origin is d, then $l d^{2}$ is equal to __________. [2024]

Ans.
(108)

Equation of focal chord

$y - 0 = \tan θ \cdot (x - 3)$

$\Rightarrow \tan θ \cdot x - y - 3 \tan θ = 0$

$∴$ Distance from origin,

$d =$ $| \frac{- 3 \tan θ}{\sqrt{1 + \tan^{2} θ}} |$

$l$ $= 4 \times 3 {cosec}^{2} θ$

$l$ $\cdot d^{2} = \frac{9 \tan^{2} θ}{1 + \tan^{2} θ} \times 12 {cosec}^{2} θ$

$= \frac{108 {cosec}^{2} θ}{1 + \cot^{2} θ} = 108$

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