JEE Main 2023 (Online) 13th April Evening ShiftQuestion 1
Let $(\alpha,\beta)$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then $\alpha+2\beta$ and $2\alpha-\beta$ are the roots of the equation
Points and Straight Lines
Solution
Vertices: $15x-y=82$ & $6x-5y=-4$ give $(6,8)$; $6x-5y=-4$ & $9x+4y=17$ give $(1,2)$; $15x-y=82$ & $9x+4y=17$ give $(5,-7)$. Centroid $=\left(\frac{12}{3},\frac{3}{3}\right)=(4,1)$, so $\alpha=4,\beta=1$. $\alpha+2\beta=6$, $2\alpha-\beta=7$. Sum $=13$, product $=42$: $x^2-13x+42=0$. Answer (D).