JEE Main 2022 · 26 Jul · Shift 1 · Mathematics Q21

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MathematicsInteger+4 / -1Question 21 of 90

If for some $p,q,r\in\mathbf R$ , not all have same sign, one of the roots of the equation $\left(p^2+q^2\right)x^2-2q(p+r)x+q^2+r^2=0$ is also a root of the equation $x^2+2x-8=0$ , then $\dfrac{q^2+r^2}{p^2}$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 272

Rewrite as $(px-q)^2+(qx-r)^2=0$ : a real root needs $x=\frac qp=\frac rq$ , a root of $x^2+2x-8$ : 2 or $-4$ .

Mixed signs rule out $x=2$ , so $x=-4$ : $q=-4p$ , $r=16p$ , and $\frac{q^2+r^2}{p^2}=272$ .

Answer: 272

Topics:Nature of roots Roots and coefficients

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JEE Main 2022 · 26 Jul · Shift 1, Mathematics question 21