If p and q are the lengths of the perpendiculars from the origin on the lines, x - y = k2 …

Question

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x\csc\alpha - y\sec\alpha = k\cot2\alpha$ and $x\sin\alpha+y\cos\alpha=k\sin2\alpha$ respectively, then $k^2$ is equal to:

Accepted Answer

$$p = \frac{|k\cot2\alpha|}{\sqrt{\csc^2\alpha+\sec^2\alpha}}, \qquad q=\frac{|k\sin2\alpha|}{\sqrt{\sin^2\alpha+\cos^2\alpha}}=|k\sin2\alpha|$$

Simplifying $p$: $\csc^2\alpha+\sec^2\alpha = \dfrac1{\sin^2\alpha\cos^2\alpha}$, so
$$p = k\cot2\alpha\cdot\sin\alpha\cos\alpha = k\cdot\frac{\cos2\alpha}{\sin2\alpha}\cdot\frac{\sin2\alpha}2 = \frac{k\cos2\alpha}2$$

So $\cos2\alpha = \dfrac{2p}k$ and $\sin2\alpha=\dfrac qk$:
$$\sin^22\alpha+\cos^22\alpha=1 \implies \frac{q^2}{k^2}+\frac{4p^2}{k^2}=1 \implies 4p^2+q^2=k^2$$

Solutions · 1