Quadratic Equations
Roots Difference Condition — Sum of λ Values
nta_pyq_2026_jan
Grade 11
Question:
If $\alpha,\beta$, where $\alpha<\beta$, are the roots of the equation $\lambda x^2-(\lambda+3)x+3=0$ such that $\dfrac{1}{\alpha}-\dfrac{1}{\beta}=\dfrac{1}{3}$, then the sum of all possible values of $\lambda$ is
Step-by-Step Solution
Key Concept: $\alpha+\beta=\tfrac{\lambda+3}{\lambda}$, $\alpha\beta=\tfrac{3}{\lambda}$. $\tfrac{1}{\alpha}-\tfrac{1}{\beta}=\tfrac{\beta-\alpha}{\alpha\beta}=\tfrac{1}{3}\Rightarrow\beta-\alpha=\tfrac{\alpha\beta}{3}=\tfrac{1}{\lambda}$.
$\lambda=2$ or $4$. Sum $=6$.
Correct Answer: 2