Question

A quadratic equation \(x^2+bx+c=0\) has two real roots. If the difference between the reciprocals of the roots is \(\frac{1}{3}\) , and the sum of the reciprocals of the squares of the roots is \(\frac{5}{9}\) , then the largest possible value of \((b+c)\) is

• A. 7
• B. 8
• C. 9
• D. None of Above

Answer 1

The Correct Option is C

Let the roots of the quadratic equation \(x^2 + bx + c = 0\) be \(\alpha\) and \(\beta\). 

Given:

• \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{5}{9}\)
• \(\left( \frac{1}{\alpha} - \frac{1}{\beta} \right)^2 = \frac{1}{9}\)

Step 1: Use identity

We know: \[ \left( \frac{1}{\alpha} - \frac{1}{\beta} \right)^2 = \frac{1}{\alpha^2} + \frac{1}{\beta^2} - \frac{2}{\alpha\beta} \] Substituting the given values: \[ \frac{1}{9} = \frac{5}{9} - \frac{2}{\alpha\beta} \] Rearranging: \[ \frac{2}{\alpha\beta} = \frac{4}{9} \Rightarrow \alpha\beta = \frac{9}{2} \]

Step 2: Express \(\alpha^2 + \beta^2\) in terms of \(\alpha + \beta\) and \(\alpha\beta\)

We use the identity: \[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{\alpha^2 \beta^2} \] From the given: \[ \frac{5}{9} = \frac{\alpha^2 + \beta^2}{(\alpha\beta)^2} \] \[ \Rightarrow \alpha^2 + \beta^2 = \frac{5}{9} \cdot \left( \frac{81}{4} \right) = \frac{405}{36} = \frac{45}{4} \]

Step 3: Find \((\alpha + \beta)^2\)

Using identity: \[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] Substituting known values: \[ \frac{45}{4} = (\alpha + \beta)^2 - 2 \cdot \frac{9}{2} = (\alpha + \beta)^2 - 9 \] \[ \Rightarrow (\alpha + \beta)^2 = \frac{45}{4} + 9 = \frac{81}{4} \Rightarrow \alpha + \beta = \pm \frac{9}{2} = \pm 4.5 \]

Step 4: Relate to coefficients \(b\) and \(c\)

From standard form: \[ \alpha + \beta = -b, \quad \alpha\beta = c \] So: \[ b = -(\alpha + \beta) = \mp 4.5, \quad c = \frac{9}{2} = 4.5 \]

Step 5: Find maximum value of \(b + c\)

Consider: \[ b + c = -(\alpha + \beta) + \alpha\beta \] To maximize \(b + c\), take \(\alpha + \beta = -4.5\), so: \[ b = 4.5, \quad c = 4.5 \Rightarrow b + c = 4.5 + 4.5 = \boxed{9} \]

Final Answer: Option (C): 9