Question

The ends of a rod of length $r$ move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio $1:1$, is

• A. $4x^2+4y^2=r^2$
• B. $x^2+y^2=r^2$
• C. $4x^2+4y^2=3r^2$
• D. $2(x^2+y^2)=r^2$

Answer 1

The correct option is A $4x^2+4y^2=r^2$

Let the two mutually perpendicular lines be coordinate axes,

Let our point be (x, y)

Let the points at which rod touches the coordinate axes be $(x_1,0)$ and $(0,y_1)$

Then according to Pythagoras theorem

$x_1^2+y_1^2=r^2$ let us name this equation 1

Also using

$x=(m{x}_1+n{x}_2)/(m+n)$

We have

$y_1=2y$

$x_1=2x$,

Using putting values in equation 1 we have,

$(2x{)}^2+(2y{)}^2=r^2$

$4x^2+4x^2=r^2$