IIT-JEE 2001 ScreeningQuestion 1
Let $\alpha,\beta$ be the roots of $x^{2}-x+p=0$ and $\gamma,\delta$ be the roots of $x^{2}-4x+q=0$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then the integral values of $p$ and $q$ respectively, are
Quadratic Equation · Relation Between Roots and Coefficients
Solution
Concept. Write the four numbers as $\alpha,\alpha r,\alpha r^2,\alpha r^3$; the two sums give $r$, then the products give $p,q$. Formulas used. (1) Sum and product of rootsStep 1 - Sums. $\alpha+\alpha r=\alpha(1+r)=1$ and $\alpha r^2+\alpha r^3=\alpha r^2(1+r)=4$. Dividing: $r^2=4$. Step 2 - $r=2$. $\alpha=\frac13$ and $p=\alpha\cdot\alpha r=\frac29$, not an integer. Rejected. Step 3 - $r=-2$. $\alpha=\frac{1}{1-2}=-1$. Then $$p=\alpha^2r=-2,\qquad q=\alpha r^2\cdot\alpha r^3=\alpha^2r^5=-32.$$ Answer: (A)