If the vectors ` hat i-2x hat j+3y hat k\ a n d\ 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

Explanation

Let, `vec a = hat I – 2 x hat j + 3 y hat k` and `vec b = hat i + 2 x hatj – 3 y hat k`

It is given that the vectors are perpendicular. So, their dot product is zero.

`vec a * vec b = 0`

`Rightarrow (hat I – 2 x hat j + 3 y hat k)*(hat I + 2 x hat j – 3 y hat k ) = 0`

`Rightarrow 1 – 4x^2 – 9y^2 = 0`

`Rightarrow 4x^2 + 9y^2 = 1`

Dividing both sides by `36`, we get

`x^2/9 + y^2/4 = 1`

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