Question

Consider three quadratic equations, x² + ax + bc = 0, x² + bx + ca = 0, x² + cx + ab = 0. If each pair of these equations has a common root, then the sum and product of these common roots are:
1. \(\rm Sum=\dfrac{-(a+b+c)}{2},\ Product=a^2b^2c^2\)
2. \(\rm Sum=\dfrac{-(a+b+c)}{2},\ Product=abc\)
3. \(\rm Sum=\dfrac{(a+b+c)}{2},\ Product=a^2b^2c^2\)
4. \(\rm Sum=\dfrac{(a+b+c)}{2},\ Product=2abc\)

Answer 1

Correct Answer - Option 2 : \(\rm Sum=\dfrac{-(a+b+c)}{2},\ Product=abc\)

Given:

Three quadratic equations, x2 + ax + bc = 0, x2 + bx + ca = 0, x2 + cx + ab = 0

Concept used:

A general quadratic equation is ax² + bx + c = 0

If α and β is the roots of the equation then,

α × β = c/a and α + β = -b/a

Calculation:

Let α and β be the roots of the equation x2 + ax + bc = 0

⇒ α + β = -a.......(1)

⇒ α β = bc........(2)

And β & z be the roots of the equation x2 + bx + ca = 0

⇒ β + z = -b.......(3)

⇒ βz = ac........(4)

And α & z be the roots of the equation x2 + cx + ab = 0

⇒ α + z = -c......(5)

⇒ α β = ab.......(6)

Adding (1),(2) & (5), we get 2α + 2β + 2z = -a - b - c

⇒ \(α + β + z = \frac{-(a + b + c)}{2} \) 

⇒ Multiplying (2),(4) & (6), we get α² β² z² = a²b²c²

⇒ α β z = abc

∴ \(\rm Sum=\dfrac{-(a+b+c)}{2},\ Product=abc\)