Variable complex number z satisfies the equation `|z-1+2i|+|z+3-i|=10`. Prove that locus of complex number z is ellipse. Also, find the centre, foci and eccentricity of the ellipse.

Explanation

Given equation is `|z-(1-2i)|+|z-(-3+i)|=10`
`|z-(1-+i)|`= distacne between z and 1-2i`(=z+(1))`
`|z-(-3_(i))|`= distacne between z and-+i`(=z_(2))`
Thus, sum of distance of z form points (1-2i) and (-3+i) is constant 10. So, locus of z is ellips e with foci at `z_(1) and z_(2)`.
Centre is midpoint of `z_(1) and z_(2)`, which is `-1-(i)/(2)`
Distane between foci = `|z_(1)-z_(2)|-|1-2i-(-3+i)|=|4-3i|=5 = 2ae`
Major axis =10=2a
`:.` Eceentricity `=e=(5)/(10)=(1)/(2)`