Question

A point $(\alpha ,\beta )$ lies on a circle $x^2+y^2=1$, then locus of the point $(3\alpha +2\beta )$ is a$/$an.

• A. Straight line
• B. Ellipse
• C. Parabola
• D. None of these

Answer 1

The correct option is B Ellipse

Point will be $(3\alpha ,2\beta )$ not $(3\alpha +2\beta )$

Now $x^2+y^2=1$

Radium is $1$ unit,hence parametric co - ordinate is

$(\alpha ,\beta )=(1\cos \theta ,1\sin \theta )=(\cos \theta ,\sin \theta )$

Hence

Point is $(3\cos \theta ,2\sin \theta )$

Hence

$(x,y)=(3\cos \theta ,2\sin \theta )$

$x=3\cos \theta$

$\Rightarrow \frac{x}{3}\cos \theta$ ...(i)

$y=2\sin \theta$

$\Rightarrow \frac{y}{2}=\sin \theta$ ...(ii)

$(i{)}^2+(ii{)}^2$

$\frac{x^2}{9}+\frac{y^2}{4}{\cos }^2\theta +{\sin }^2\theta$

$\frac{x^2}{9}+\frac{y^2}{4}=1$

which is equation of ellipse