Question

A variable line whose slope is $-2$ cuts the $x$ and $y$ axes respectively at points $A$ and $C$. A rhombus $ABCD$ is completed such that vertex $B$ lies on the line $y=x$. Then the locus of vertex $D$ is ?

• A. $2x+y=1$
• B. $x-y=0$
• C. $x+y=0$
• D. $x+2y=0$

Answer 1

The correct option is C $x+y=0$

Given,

$y=-2x+c$

$AB=AD=BC=CD$

$y=x$

$y-\frac{c}{2}=\frac{1}{2}(x-\frac{c}{4})$

$2y=x+\frac{3c}{4}$

$\Rightarrow x=y=\frac{3c}{4}$

$\frac{\alpha +\frac{3c}{4}}{4}=\frac{c}{4}$

$\alpha =-\frac{c}{4}$

$\frac{\beta +\frac{3c}{4}}{4}=\frac{c}{2}$

$\beta =\frac{c}{4}$

$\alpha =-\beta$

$x=-y$

$\Rightarrow x+y=0$