If `a ,b ,c` are distinct positive numbers, then the nature of roots of the equation `1//(x-a)+1//(x-b)+1//(x-c)=1//x` is all real and is distinct all real and at least two are distinct at least two real d. all non-real

Options

A. all real and distinct
B. all real and at least two are distinct
C. al least two real
D. all non-real

Answer

1

Explanation

The equation on simplifying gives
`x(x - b) (x - c) + x (x - c) (x - a) + x(x - a)(x - b)`
`- (x - a) (x - b)(x-c) = 0` (1)
Let ` f(x) = x(x - b) (x - c) + x(x - c) (x -a) + x (x - a)( x - b)`
` - (x - a) (x - b) (x - c)`
We can assume without loss of generality that a ` lt a lt c`. Now
` f(a) = a (a - b) (a - c) gt 0 `
` f(b) = b (b - c) (b - a) lt 0 `
`rArr f(c) = c (c - a) (c - b) gt 0`
So, one root of (1) lies in (a, b) and one root in (b,c) Obiviously
the third root must also be real .