Find the locus of a point such that the sum of its distances from the points `(0,2)a n d(0,-2)` is`6.`

Explanation

The locus of points, whose sum of distance from two fixed points is constant is an ellipse.

For ellipse. `frac{x^2}{b^2}+frac{y^2}{a^2}=1`

Here, the constant sum `2a=6` or `a=3`

And distance between focii `(0,2)` and `(0,−2)` is `4`.

`2ae=4 or e= frac{2}{3}`

Since the focii lie on the y axis, the equation of the ellipse will be of the form `frac{x^2}{a^2(1-e^2)}+frac{y^2}{a^2}=1`

`frac{x^2}{9(1-frac{4}{9})}+frac{y^2}{9}=1`

`frac{x^2}{5}+frac{y^2}{9}=1`
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