The sum of roots of equation `1/(x+a)+1/(x+b)=1/c` is zero find the product of roots of equation a)0 b) `((a+b)/2)` c)`-((a^2+b^2)/2)` d) `2(a^2+b^2)`

Explanation

We have,
`(1)/(x + a) + (1)/(x +b) = (1)/(c)`
`rArr x^(2) + (a + b - 2c)x + (ab - bc -ca) = 0`
Let `alpha, beta` be the roots of this equation. Then,
`alpha+ beta = -(a + b - 2c) and alpha beta = ab - bc - ca`
It is given that
`alpha + beta = 0`
`rArr -(a+b-2c) = 0`
or `c=(a+b)/(2)` (1)
`therefore alpha beta = ab - bc - ca `
= ab -c (a + B)
`ab - ((a+b)/(2)) (a + b) [Using (1)]`
`=(2ab - (a+ b)^(2))/(2) = -(1)/(2) (a^(2) + b^(2))`