If the vectors $\mathbf{i}-2 x \mathbf{j}-3 y \mathbf{k}$ and $\mathbf{i}+3 x \mathbf{j}+2 y \mathbf{k}$ are…
If the vectors $\mathbf{i}-2 x \mathbf{j}-3 y \mathbf{k}$ and $\mathbf{i}+3 x \mathbf{j}+2 y \mathbf{k}$ are orthogonal to each other, then the locus of the point $(x, y)$ is
a circle
an ellipse
a parabola
a straight line
Solution
Since, vectors are orthogonal.
$\begin{aligned}
& \therefore \quad(\mathbf{i}-2 x \mathbf{j}-3 y \mathbf{k}) \cdot(\mathbf{i}+3 x \mathbf{j}+2 y \mathbf{k})=0 \\
& \Rightarrow \quad 1-6 x^2-6 y^2=0 \Rightarrow x^2+y^2=\frac{1}{6}
\end{aligned}$
Hence, the locus of a point is a circle.