Question

The locus of the point of intersection of the lines x cosθ + y sinθ = a and x sinθ – y cosθ = b is a circle of radius __________.

Answer 1

Let P = (x, y) be the point of intersection of lines

$$\begin{gathered} x\cos \theta +y\sin \theta =a...\left(1\right)\mathrm{and} \\ x\sin \theta -y\cos \theta =b...\left(2\right) \\ \mathrm{By}\mathrm{squaring}\mathrm{and}\mathrm{adding}\mathrm{equation}\left(1\right)\mathrm{and}\left(2\right), \\ \mathrm{we}\mathrm{get}, \\ {\left(x\cos \theta +y\sin \theta \right)}^2+\left(x\sin \theta -y\cos \theta \right)=a^2+b^2 \\ \mathrm{i}.\mathrm{e}.x^2{\cos }^2\theta +y^2{\sin }^2\theta +2xy\cos \theta \sin \theta +x^2{\sin }^2\theta +y^2{\cos }^2\theta -2xy\sin \theta \cos \theta =a^2+b^2 \\ \mathrm{i}.\mathrm{e}.x^2\left({\cos }^2\theta +{\sin }^2\theta \right)+y^2\left({\sin }^2\theta +{\cos }^2\theta \right)=a^2+b^2 \\ \mathrm{i}.\mathrm{e}.x^2+y^2=a^2+b^2 \\ \mathrm{i}.\mathrm{e}.\mathrm{radius}\mathrm{of}\mathrm{circle}\mathrm{formed}\mathrm{is}\sqrt{a^2+b^2} \end{gathered}$$