If $\mathrm{d}_1, \mathrm{~d}_2, \mathrm{~d}_3$ are the distances of the point $(1,2,3)$ from the…
If $\mathrm{d}_1, \mathrm{~d}_2, \mathrm{~d}_3$ are the distances of the point $(1,2,3)$ from the $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$-coordinate axes respectively then $2 d_2^2+d_3^2+1=$
$d_1^2$
$2 d_1^2$
$3 d_1^2$
$4 d_1^2$
Solution
We have distance $\mathrm{d}_1, \mathrm{~d}_2$ and $\mathrm{d}_3$ of the point $(1.2,3)$ from X,Y, Z.
$\begin{aligned} & \Rightarrow \mathrm{d}_1=\sqrt{2^2+3^2}=\sqrt{13} \\ & \mathrm{~d}_2=\sqrt{1^2+3^2}=\sqrt{10} \\ & \mathrm{~d}_3=\sqrt{1^2+2^2}=\sqrt{5}\end{aligned}$
Now $2 \mathrm{~d}_2^2+\mathrm{d}_3^2+1=2 \times 10+5+1=26$
$=2 \mathrm{~d}_1^2$