Let alpha, beta be the roots of the equation x^2 - px + r = 0 and (alpha)/(2), 2beta be the roots of the equation x^2 - qx + r = 0 . Then the value of r is

Quadratic Equation: Let alpha, beta be the roots of the equation x^2 - px + r = 0 and… | JEE Advanced 2007 Paper 1

JEE Advanced 2007 Paper 1Question 1

Let $\alpha, \beta$ be the roots of the equation $x^2 - px + r = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 - qx + r = 0$. Then the value of $r$ is

Quadratic Equation · Relation Between Roots and Coefficients

$\frac{2}{9}(p-q)(2q-p)$$\frac{2}{9}(q-p)(2p-q)$$\frac{2}{9}(q-2p)(2q-p)$$\frac{2}{9}(2p-q)(2q-p)$✓ Correct answer

Solution

Concept. Both equations have product $r$; the two sums give $\alpha$ and $\beta$. Formulas used. (1) Sum and product of rootsStep 1 - Sums. $\alpha+\beta=p$ and $\frac\alpha2+2\beta=q$. Step 2 - Solve. Put $\alpha=p-\beta$: $\frac{p}{2}+\frac{3\beta}{2}=q$, so $$\beta=\frac{2q-p}{3},\qquad\alpha=\frac{4p-2q}{3}=\frac{2(2p-q)}{3}.$$ Step 3 - Product. $r=\alpha\beta=\dfrac29(2p-q)(2q-p)$. Answer: (D)

Quadratic Equation - JEE Mathematics question (JEE Advanced 2007 Paper 1)

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