Question

If a, b be the roots of x² – px + q = 0 and a¢ , b¢ be the roots of x² – p¢x + q¢ = 0 then the value of (a – a¢)² + (b – a¢)² + (a  – b¢)² + (b – b¢)² is –

• A. 2{p² – 2q + p¢² – 2q¢ – pp¢}
• B. 2{p² – 2q + p¢² – 2q¢ + qq¢}
• C. 2{p² – 2q – p¢² – 2q¢ + pp¢}
• D. 2{p² – 2q – p¢² – 2q¢ – qq¢}

Answer 1

Answer: (A)

2{p² – 2q + p¢² – 2q¢ – pp¢}

Answer 2

Since a, b are the roots of x² – px + q = 0 and a¢ , b¢ are the roots of x² – p¢x + q¢ = 0 then

a + b = P, ab = q, a¢  + b¢ = p¢, a¢b¢ = q¢

Now, (a – a¢)² + (b – a¢)² + (a  – b¢)² + (b – b¢)²

= 2(a² + b¢) + 2(a¢² + b¢²) – 2 a¢  (a + b) – 2b¢ (a + b)

= 2{(a + b)² – 2ab + (a¢  + b¢)² – 2a¢b¢ –

a + b) (a¢ + b¢)} = 2(p² – 2q + p¢² – 2q¢ – pp¢)