If the sum of the roots of the quadratic equation `1/(x+p)+1/(x+q)=1/r` is zero, show that the product of the roots is `(-(p^(2)+q^(2))/2)`

Explanation

`1/(x+p)+1/(x+q)=1/r`
`:.((x+q)+(x+p))/((x+p)(x+q))=1/r :.(2x+p+q)/((x+p)(x+q))=1/r`
`:.r(2x+p+q)=(x+p)(x+q)`
`:.(x+p)(x+q)=r(2x+p+q)`
`:.x^(2)+x(p+q)+pq=2rx+rp+rq`
`:.x^(2)+ul(x(p+q))=ul(pq)-ul(2rx)-ul(rp-rq)=0`
`:.x^(2)+x(p+q-2r)[pq-r(p+q)]=0`
Comparing with `ax^(2)+bx+c=0, a=1, b=p+q-2r` and `c=pq-r(p+q)`
`:.` sum of the roots `=alpha+beta=-b/a=-(p+q-2r)`
and the product of the roots `= alpha beta=c/a =pq-r(p+q)......... ( :' a=1)`
Sum of the roots `=0` ............(Given)
`:.-(p+q-2r)=0 :.p+q-2r=0`
`:.p+q=2r:.(p+q)/2=r`
Substituting the value of `r` in `pq-r(p+q)`, the product of the roots,
`pq-((p+q)/2)(p+q)`
`=(2pq-(p+q)^(2))/2=(-p^(2)-q^(2))/2=-(p^(2)+q^(2))/2`