JEE MAIN - Mathematics (2021 - 31st August Morning Shift - No. 8)
If p and q are the lengths of the perpendiculars from the origin on the lines,
x cosec $$\alpha$$ $$-$$ y sec $$\alpha$$ = k cot 2$$\alpha$$ and
x sin$$\alpha$$ + y cos$$\alpha$$ = k sin2$$\alpha$$
respectively, then k2 is equal to :
x cosec $$\alpha$$ $$-$$ y sec $$\alpha$$ = k cot 2$$\alpha$$ and
x sin$$\alpha$$ + y cos$$\alpha$$ = k sin2$$\alpha$$
respectively, then k2 is equal to :
4p2 + q2
2p2 + q2
p2 + 2q2
p2 + 4q2
Explanation
First line is $${x \over {\sin \alpha }} - {y \over {\cos \alpha }} = {{k\cos 2\alpha } \over {\sin 2\alpha }}$$
$$ \Rightarrow x\cos \alpha - y\sin \alpha = {k \over 2}\cos 2\alpha $$
$$ \Rightarrow p = \left| {{k \over 2}\cos \alpha } \right| \Rightarrow 2p = \left| {k\cos 2\alpha } \right|$$ .... (i)
second line is $$x\sin \alpha + y\cos \alpha = k\sin 2\alpha $$
$$ \Rightarrow q = \left| {k\sin 2\alpha } \right|$$ .... (ii)
Hence, $$4{p^2} + {q^2} = {k^2}$$ (From (i) & (ii))
$$ \Rightarrow x\cos \alpha - y\sin \alpha = {k \over 2}\cos 2\alpha $$
$$ \Rightarrow p = \left| {{k \over 2}\cos \alpha } \right| \Rightarrow 2p = \left| {k\cos 2\alpha } \right|$$ .... (i)
second line is $$x\sin \alpha + y\cos \alpha = k\sin 2\alpha $$
$$ \Rightarrow q = \left| {k\sin 2\alpha } \right|$$ .... (ii)
Hence, $$4{p^2} + {q^2} = {k^2}$$ (From (i) & (ii))
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