Question

If the vectors $\hat{i}-2x\hat{j}+3y\hat{k}\mathrm{and}\hat{i}+2x\hat{j}-3y\hat{k}$ are perpendicular, then the locus of (x, y) is
(a) a circle
(b) an ellipse
(c) a hyperbola
(d) None of these

Answer 1

(b) an ellipse

$$\begin{gathered} \mathrm{Let},\overrightarrow{a}=\hat{i}-2x\hat{j}+3y\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}+2x\hat{j}-3y\hat{k} \\ \text{It is}\text{given that the vectors are perpendicular. So, their dot product is zero.} \\ \overrightarrow{a}.\overrightarrow{b}=0 \\ \Rightarrow \left(\hat{i}-2x\hat{j}+3y\hat{k}\right).\left(\hat{i}+2x\hat{j}-3y\hat{k}\right)=0 \\ \Rightarrow 1-4x^2-9y^2=0 \\ \Rightarrow 4x^2+9y^2=1 \\ \text{Dividing both sides by 36, we get} \\ \frac{x^2}{9}+\frac{y^2}{4}=1 \\ \text{This is an ellipse.} \end{gathered}$$