Question

If the sum of the roots of the equation 1/(x+a)+1/(x+b)=1/c is zero, then find its product of the roots.

Answer 1

product of roots is -(a² + b²)/2

given, sum of the roots of the equation,

1/(x + a) + 1/(x + b) = 1/c is zero.

first resolve the equation,

1/(x + a) + 1/(x + b) = 1/c

⇒(2x + a + b)/(x + a)(x + b) = 1/c

⇒2cx + ac + bc = x² + (a + b)x + ab

⇒x² + (a + b - 2c)x + (ab - bc - ca) = 0

now sum of roots = -coefficient of x/coefficient of x²

⇒0 = -(a + b - 2c)

⇒a + b = 2c

⇒c = (a + b)/2 ........(1)

now product of roots = constant/coefficient of x²

= (ab - bc - ca)

= ab - c(b + a)

= ab - (a + b)²/2 [ from eq (1)]

= (2ab - a² - b² - 2ab)/2

= -(a² + b²)/2