Let p, q(p > q) be the roots of the quadratic equation x2 + bx + c = 0 where c > 0. If p2 + q2 − 11pq = 0, then what is p − q equal to ?

Concept:

If α and β are the roots of the equation ax² + bx + c = 0,

then sum of roots = α + β = -\(b \over a\)

and product of roots = αβ = \(c \over a\)

Calculation:

If p, q(p > q) are the roots of the quadratic equation x2 + bx + c = 0 

then, p + q = -b  __(i)

and pq = c   __(ii)

Squaring equation (i),

⇒ (p + q)² = (-b)²

⇒ p² + q² + 2pq = b²

⇒ p2 + q2 + 2c = b2  (From (i))

⇒ p2 + q2 = b2 - 2c  __(iii)

Take equation (iii) - 11(ii),

⇒ p2 + q2 − 11pq = b2 - 2c - 11c = 0

⇒ b² = 13c  __(iv)

Now (p - q)² =  p2 + q2 - 2pq

Put values from (ii) and (iii),

⇒ (p - q)2 =  b2 - 2c  - 2c = b2 - 4c

⇒ (p - q)2 =  13c - 4c {from (iv)}

⇒ (p - q)2 =  9c

⇒ p - q = 3√c

∴ The correct answer is option (1).