Two positive charges `q_(1)` and `q_(2)` are located at the points with redius vectors `r_(1)` and `r_(2)`. Find a negative charge `q_(3)` and a radius vector `r_(3)` of the point at which it has to be placed for the force acting on each of the three charges to be equal to zero.

Explanation

Let us choose coordinate area as shown in the figure and fix three charges, `q_(1), q_(2)` and `q_(3)` having position vectors `vec(r_(1)), vec(r_(2))` and `vec(r_(3))` respectively,
Now, for the equilibrium of `q_(3)`
`(+ q_(2) q_(3) (vec(r_(2)) - vec(r_(3))))/(|vec(r_(2)) - vec(r_(3))|^(3)) + (q_(1) q_(3) (vec(r_(1)) - vec(r_(3))))/(|vec(r_(1)) - vec(r_(3))|^(3)) = 0`

`(q_(2))/(| vec(r_(2)) - vec(r_(3))|^(2)) = (q_(1))/(| vec(r_(2)) - vec(r_(3))|^(2))`
because `(vec(r_(2)) - vec(r_(3)))/(|vec(r_(2)) - vec(r_(3))|) = - (vec(r_(1)) - vec(r_(3)))/(|vec(r_(1)) - vec(r_(3))|)`
or, `sqrt(q_(2)) (vec(r_(1)) - vec(r_(3))) = sqrt(q_(1)) (vec(r_(3)) - vec(r_(2)))`
or, `vec(r_(3)) = (sqrt(q_(2)) vec(r_(1)) + sqrt(q_(1)) vec(r_(2)))/(sqrt(q_(1)) + sqrt(q_(2)))`
Also for the equilibriumk of `q_(1)`,
`(q_(3) (vec(r_(3)) - vec(r_(1))))/(|vec(r_(3)) - vec(r_(1))|^(3)) + (q_(2) (vec(r_(2)) - vec(r_(1))))/(|vec(r_(2)) - vec(r_(1))|^(3)) = 0`
or, `q_(3) = (-q_(2))/(|vec(r_(2)) - vec(r_(1))|^(2)) |vec(r_(1)) - vec(r_(3))|^(2)`
Substiuting the value of `vec(r_(3))`, we get,
`q_(3) = (-q_(1) q_(2))/((sqrt(q_(1)) + sqrt(q_(2)))^(2))`