Verified Solution • 2025–26 Session moderate

QJEEM21Maf43a: JEE Main 2021 Mathematics - Class 11 JEEM21Maf43a Solution: If p and q are the lengths of the perpendiculars from the…

If p and q are the lengths of the perpendiculars from the origin on the lines,

x cosec α\alpha −- y sec α\alpha = k cot 2 α\alpha and

x sin α\alpha + y cos α\alpha = k sin2 α\alpha

respectively, then k2 is equal to :

Multiple Choice Options

  • A 4p2 + q2
  • B 2p2 + q2
  • C p2 + 2q2
  • D p2 + 4q2

Step-by-Step Verified Solution

First line is xsin⁡α−ycos⁡α=kcos⁡2αsin⁡2α{x \over {\sin \alpha }} - {y \over {\cos \alpha }} = {{k\cos 2\alpha } \over {\sin 2\alpha }}

⇒xcos⁡α−ysin⁡α=k2cos⁡2α\Rightarrow x\cos \alpha - y\sin \alpha = {k \over 2}\cos 2\alpha

⇒p=∣k2cos⁡α∣⇒2p=∣kcos⁡2α∣\Rightarrow p = \left| {{k \over 2}\cos \alpha } \right| \Rightarrow 2p = \left| {k\cos 2\alpha } \right| .... (i)

second line is xsin⁡α+ycos⁡α=ksin⁡2αx\sin \alpha + y\cos \alpha = k\sin 2\alpha

⇒q=∣ksin⁡2α∣\Rightarrow q = \left| {k\sin 2\alpha } \right| .... (ii)

Hence, 4p2+q2=k24{p^2} + {q^2} = {k^2} (From (i) & (ii))

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