If the vectors $\hat{i}-2x\hat{j}+3y\hat{k}$ and $\hat{i}+2x\hat{j}-3y\hat{k}$ are perpendicular, then the locus of (x, y) is :

an ellipse

The correct answer is Option (2) → an ellipse

We are given two vectors,

$A=\hat{i}-2x\hat{j}+3y\hat{k}$

$B=\hat{i}+2x\hat{j}-3y\hat{k}$

if the vectors are perpendicular,

$A.B=0$

$A.B=(1)(1)+(-2x)(2x)+(3y)(-3y)$

$=1-4x^2-9y^2$

∴ it is an ellipse.