JEE Main 2022 (Online) 28th July Morning ShiftQuestion 1
For $p,q\in\mathbf{R}$, consider the real valued function $f(x)=(x-p)^2-q$, $x\in\mathbf{R}$ and $q>0$. Let $a_1,a_2,a_3$ and $a_4$ be in an arithmetic progression with mean $p$ and positive common difference. If $|f(a_i)|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is ________.
Quadratic Equation · Relation Between Roots and Coefficients
Answer$50$
Solution
Concept. Write the A.P. as $p\pm d$, $p\pm3d$; then $f$ takes only two values on it. Formulas used. (1) $f(x)=(x-p)^2-q$ has roots $p\pm\sqrt q$Step 1 - Values of $f$. $a_i-p=-3d,-d,d,3d$ ($d>0$), so $f(a_i)$ is $d^2-q$ or $9d^2-q$. Step 2 - Signs. Both have absolute value $500$. They cannot be equal ($8d^2\ne0$), so $$d^2-q=-500,\qquad9d^2-q=500.$$ Step 3 - Solve. $8d^2=1000$, $d^2=125$, $q=625$. Step 4 - Difference of roots. By (1), $2\sqrt q=50$. Answer: 50