Question

If vectors $(\hat{i}-2x\hat{j}-3y\hat{k})$ and $(\hat{i}+3x\hat{j}+2y\hat{k})$ are orthogonal to each other, then the locus of the point $(x,y)$ is

• A. a straight line
• B. a parabola
• C. a circle
• D. an ellipse

Answer 1

The correct option is C a circle

Let $\overrightarrow{a}=(\hat{i}-2x\hat{j}-3y\hat{k})$ and $\overrightarrow{b}=(\hat{i}+3x\hat{j}+2y\hat{k})$
Given $\overrightarrow{a}$ and $\overrightarrow{b}$ are orthogonal.
$\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=0$
$\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=1-6x^2-6y^2=0$
$\Rightarrow x^2+y^2=\frac{1}{6}$
$\therefore$ Locus of the point $(x,y)$ is a circle