If the roots of the equation `a(b-c)x^2+b(c-a)x+c(a-b)=0` are equal, show that `2//b=1//a+1//cdot`

Explanation

Since the roots of the given equations are equal, its discriminant is zero, i. e.,
`b^(2)(c - a)^(2) - 4a(b - c) c(a - b) = 0`
or `b^(2) (c^(2) + a^(2) - 2ac) - 4ac (ba - ca -b^(2) + bc) = 0`
or `a^(2)b^(2) +b^(2)c^(2) + 4a^(2) c^(2) + 2b^(2)ac - 4a^(2) bc- 4abc^(2) = 0`
`(ab + bc- 2ac)^(2) = 0 `
or `ab + bc -2ac = 0`
or `ab + bc = 2ac`
or `(1)/(c)+(1)/(a)=(2)/(b)` [dividing both sides byu abc]
or `(2)/(b)=(1)/(a)+(1)/(c)`