Question

A line segment $AB$ of length $2$ moves with its ends on the axes. The locus of the point $P$ which divides the segment in the ratio $1:1$ is

• A. $9x^2+4y^2=1$
• B. $x^2+y^2=1$
• C. $x^2+4y^2=8$
• D. $9x^2+9y^2=4$

Answer 1

The correct option is B $x^2+y^2=1$

Let (x, y) be a point which divides in $1:1$ ratio

$\therefore$ coordinates of point $A=(2x,0)$

coordinates of point $B=(0,2y)$

because

$x=\frac{m\left(x_A\right)+n\left(x_B\right)}{m+n}$

$\therefore x=\frac{x_A}{2}$

$\therefore x_A=2x$

Similarly $y_B=2y$

$\therefore (2x-0{)}^2+(0-2y{)}^2=4$

$\therefore 4x^2+4y^2=4$

$\therefore x^2+y^2=1$.