Question

The locus represented by the point $(x,y)$ which coordinates $x=5\cos \theta +3\sin \theta$ & $y=5\sin \theta -3\cos \theta$ is

• A. $x^2-y^2=34$
• B. $x^2+y^2=16$
• C. $x^2+y^2=34$
• D. $noneofthese$

Answer 1

The correct option is C $x^2+y^2=34$

$x=5\left(\cos \theta +3sin\theta \right)$

$x^2=25{\cos }^2\theta +9{\sin }^2\theta +30\sin \theta \cos \theta -\left(1\right)$

$y=5\sin \theta -3\cos \theta$

$y^2=25{\sin }^2\theta +9{\cos }^2\theta -30\sin \theta \cos \theta -\left(2\right)$

$Adding1\&2$

$x^2+y^2=25\left({\sin }^2\theta +{\cos }^2\theta \right)+9\left({\sin }^2\theta +{\cos }^2\theta \right)$

$x^2+y^2=25+9$

$x^2+y^2=34$

Hence locus of the point $(x,y)$ is $x^2+y^2=34$