Question

The locus of the point of intersection of the lines $x\cos \alpha +y\sin \alpha =a$ and $x\sin \alpha -y\cos \alpha =b$ where $\alpha$ is variable is -

• A. $x^2+y^2=a^2+b^2$
• B. $x^2+y^2=a^2-b^2$
• C. $x^2-y^2=a^2-b^2$
• D. $x^2-y^2=a^2+b^2$

Answer 1

Answer: (A)

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

Given, $x\cos \alpha +y\sin \alpha =a......\left(1\right)$
and $x\sin \alpha -y\cos \alpha =b....\left(2\right)$
Squaring and adding both the equation we get,
$x^2({\cos }^2\alpha +{\sin }^2\alpha )+y^2({\cos }^2\alpha +{\sin }^2\alpha )+2xy(\cos \alpha .\sin \alpha -\cos \alpha .\sin \alpha )=a^2+b^2$
$\Rightarrow x^2\left(1\right)+y^2\left(1\right)+xy\left(0\right)=a^2+b^2$, since ${\sin }^2\theta +{\cos }^2\theta =1$
$\Rightarrow x^2+y^2=a^2+b^2$, which is required locus.