If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines $\mathrm{x} \operatorname{cosec} \alpha-\mathrm{y} \sec \alpha=\mathrm{k} \cot 2\alpha$ and $\mathrm{x} \sin \alpha+\mathrm{y} \cos \alpha=\mathrm{k} \sin 2\alpha$ respectively, then $\mathrm{k}^{2}$ is equal to:

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Straight Lines

p

and

q

are the lengths of the perpendiculars from the origin on the lines

x cosec α - y sec α = k cot 2 α

and

x sin α + y cos α = k sin 2 α

respectively, then

k^{2}

is equal to:

(A) $p^{2} + 2 q^{2}$
(B) $4 p^{2} + q^{2}$ ✓ Correct
(C) $p^{2} + 4 q^{2}$
(D) $2 p^{2} + q^{2}$

Answer: (B)

Solution

For

x csc α - y sec α - k cot 2 α = 0

csc^{2} α + sec^{2} α = \frac{4}{sin ^{2} 2 α}

, so

p = \frac{∣ k cot 2 α ∣}{2/∣ sin 2 α ∣} = \frac{∣ k cos 2 α ∣}{2}

, giving

4 p^{2} = k^{2} cos^{2} 2 α

. For

x sin α + y cos α - k sin 2 α = 0

q = ∣ k sin 2 α ∣

, so

q^{2} = k^{2} sin^{2} 2 α

. Adding,

4 p^{2} + q^{2} = k^{2} (cos^{2} 2 α + sin^{2} 2 α) = k^{2}

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Single correct option

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