Question

If a line segment $AM=a$ moves in the plane $XOY$ remaining parallel to $OX$ so that the left end point $A$ slides along the circle $x^2+y^2=a^2$, the locus of $M$ is

• A. $x^2+y^2=4a^2$
• B. $x^2+y^2=2ax$
• C. $x^2+y^2=2ay$
• D. $x^2+y^2-2ax-2ay=0$

Answer 1

Answer: (B)

Let the coordinates of $A$ be $(x,y)$ and $M$ be $(\alpha ,\beta )$
Since $AM$ is parallel to $OX$,

$\alpha =x+a$ and $\beta =y$
$\Rightarrow x=\alpha -a$ and $y=\beta$
As $A(x,y)$ lies on the circle $x^2+y^2=a^2$, we have
$(\alpha -a{)}^2+{\beta }^2=a^2$
$\Rightarrow {\alpha }^2-2a\alpha +{\beta }^2=0$
$\Rightarrow$ locus of $M(\alpha ,\beta )$ is
$x^2+y^2=2ax$.