Quadratic Equations
Roots in G.P. — Finding (α−β)²
nta_pyq_2024_jan
Grade 11
Question:
Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$, where $p\neq0$. If $p,q$ and $r$ be the consecutive terms of a non-constant G.P. and $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{3}{4}$, then the value of $(\alpha-\beta)^2$ is:
$\dfrac{80}{9}$
9
$\dfrac{20}{3}$
8
Step-by-Step Solution
Key Concept: Let $p=A,q=AR,r=AR^2$. Equation: $x^2+Rx-R^2=0$. $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{-R}{-R^2}=\frac{1}{R}=\frac{3}{4}\Rightarrow R=\frac{4}{3}$. $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=R^2+4R^2=5R^2$.
$R=4/3$. $(\alpha-\beta)^2=5R^2=80/9$.
Correct Answer: 1