If the vectors `hat(i)-2 x hat(j)-3yhat(k) and hat(i)+3xhat(j)+2yhat(k)` are orthogonal to each other, then what is the locus of the point (x, y) ?

Options

A. A circle
B. An ellipse
C. A parabola
D. A hyperbola

Answer

A

Explanation

Let the given vectors
`hat(i) -2xhat(j)-3yhat(k) and hat(i) + 3xhat(j)+2yhat(k)`. be `vec(A) and vec(B)`
respectively, and `theta` be the angle between
them, so, `cos theta = (vec(A).vec(B))/(|vec(A)||vec(B)|)`
These are orthogonal to each other, `theta = pi//2`
so, `vec(A). vec(B)=0`
`rArr 1-(2x)(3x)-(3y)(2y)=0`
`rArr 1-6x^(2)-6y^(2)=0`
`rArr x^(2)+y^(2)=(1)/(6)`
This equation represents an equation of a circle which is the locus of the point (x, y).