Question

The locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is __________.

Answer 1

Given line x sinθ + y cosθ = P meet axes at $\left(0,\frac{P}{\cos \theta }\right)\mathrm{and}\left(\frac{P}{\sin \theta },0\right)$

$\mathrm{i}.\mathrm{e}.\mathrm{Let}A\left(\frac{9}{\sin \theta },0\right)\mathrm{and}B\left(0,\frac{P}{\cos \theta }\right)$

Let P(h, k) be the midpoint of AB

$$\begin{gathered} \mathrm{i}.\mathrm{e}.h=\frac{P}{2\sin \theta },k=\frac{P}{2\cos \theta } \\ \mathrm{i}.\mathrm{e}.\sin \theta =\frac{p}{2h},\cos \theta =\frac{p}{2k} \\ \mathrm{i}.\mathrm{e}.\sin \theta =\frac{P}{2h}\mathrm{and}\cos \theta =\frac{P}{2k} \\ \therefore {\sin }^2\theta +{\cos }^2\theta =\frac{P^2}{4h^2}+\frac{P^2}{4k^2} \\ \mathrm{i}.\mathrm{e}.1=\frac{P^2}{4h^2}+\frac{P^2}{4k^2} \\ \mathrm{i}.\mathrm{e}.\mathrm{locus}\mathrm{is}\frac{P^2}{4x^2}+\frac{P^2}{4y^2}=1 \\ \mathrm{i}.\mathrm{e}.\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{P^2} \end{gathered}$$