CAT 2023 Slot 3 QA Question 5: A quadratic equation $x^2 + bx + c = 0$ has two real roots. If the difference between the reciprocals of the roots is $\dfrac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $\df

Question

A quadratic equation $x^2 + bx + c = 0$ has two real roots. If the difference between the reciprocals of the roots is $\dfrac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $\df

Accepted Answer

Let the roots be $\alpha$ and $\beta$. Take $u = \dfrac{1}{\alpha}$ and $v = \dfrac{1}{\beta}$.

$u - v = \dfrac{1}{3}$
$u^2 + v^2 = \dfrac{5}{9}$

Now,
$(u - v)^2 = u^2 + v^2 - 2uv$
$\dfrac{1}{9} = \dfrac{5}{9} - 2uv$
$2uv = \dfrac{4}{9}$
$uv = \dfrac{2}{9}$

Therefore,
$\alpha\beta = \dfrac{1}{uv}

A quadratic equation x2+bx+c=0x^2 + bx + c = 0x2+bx+c=0 has two real roots. If the difference between the reciprocals of the roots is 13\dfrac{1}{3}31​, and the sum of the reciprocals of the squares of the roots is 59\dfrac{5}{9}95​, then the largest possible value of (b+c)(b + c)(b+c) is

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