Question

The locus of a point which in equidistant from $x$-axis and $y$-axis is

• A. The plane $x+y=0$
• B. The plane $x-y=0$
• C. The pair of planes $xy=0$
• D. none of these

Answer 1

Answer: (A), (B)

The plane $x+y=0$
The plane $x-y=0$

Let $P(x,y,z)$ by any point on the locus.

The coordinates of $M$ the foot of the perpendicularse from $P$ on $yz$ plane are $(0,y,z)$.

Distance of $P$ from x-axis is equal to the distance of $M$ from the origin which is equal to $\sqrt{y^2+z^2}$.
Similarly distance of P from y-axis is $\sqrt{x^2+z^2}$.
Now given $\sqrt{y^2+z^2}=\sqrt{x^2+z^2}$

$\Rightarrow x^2=y^2$
$\Rightarrow x\pm y=0$