If p and q are the lengths of the perpendiculars from the origin on the lines, x cosec [] …

Question

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 :

Accepted Answer

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))

Solutions · 1