Question

Find the locus of the mid-point of the portion of the line x cos $\alpha +ysin\alpha =p$ which is intercepted between the axes.

Answer 1

$Given,lineisxcosalpha+ysin\alpha =p\frac{x}{\frac{p}{cos\alpha }}+\frac{y}{\frac{p}{sin\alpha }}interceptsonxaxisis\frac{p}{cos\alpha }andy-axisis\frac{p}{sin\alpha }LetP(x,y)bethemidpointofAB.(x,y)=(\frac{p}{\frac{cos\alpha }{2},}+0\frac{p}{\frac{sin\alpha }{2}}+0)=(\frac{p}{2cos\alpha },\frac{p}{2sin\alpha })\therefore x=\frac{p}{2cos\alpha },y=\frac{p}{2sin\alpha }2cos\alpha =\frac{p}{x},2sin\alpha =\frac{p}{y}squarebothsides,4co{s}^2\alpha =\frac{p^2}{x^2}...\left(1\right)and4si{n}^2\alpha =\frac{p^2}{y^2}...\left(2\right)\left[\right(1)+(2\left)\right]4co{s}^2\alpha +4si{n}^2\alpha =\frac{p^2}{x^2}+\frac{p^2}{y^2}4=\frac{p^2(x^2+y^2)}{x^2{y}^2}4x^2{y}^2=p^2(x^2+y^2)$