Question

The locus of point of intersection of lines $x\cos \theta +y\sin \theta =a$ and $x\sin \theta -y\cos \theta =b$ is a circle of radius

Answer 1

Equation of lines are:

$x\cos \theta +y\sin \theta =a\dots \left(1\right)$

$x\sin \theta -y\cos \theta =b\dots \left(2\right)$

Squaring each equation and adding it,

$(x\cos \theta +y\sin \theta {)}^2+(x\sin \theta -y\cos \theta {)}^2=a^2+b^2$

$\Rightarrow x^2{\cos }^2\theta +y^2{\sin }^2\theta +x^2{\sin }^2\theta +y^2{\cos }^2\theta =a^2+b^2$

$\Rightarrow x^2({\cos }^2\theta +{\sin }^2\theta )+y^2({\cos }^2\theta +{\sin }^2\theta )=a^2+b^2$

$\Rightarrow x^2\times 1+y^2\times 1=a^2+b^2$

$\Rightarrow x^2+y^2=a^2+b^2$

$\therefore$ Radius of circle is $\sqrt{a^2+b^2}$

Hence, the locus of point of intersection of lines

$x\cos \theta +y\sin \theta =aandx\sin \theta -y\cos \theta =b$ is a circle of radius $\sqrt{a^2+b^2}$