JEE Main 2022 (Online) 29th July Evening ShiftQuestion 1
If the tangent to the curve $y=x^3-x^2+x$ at the point $(a,b)$ is also tangent to the curve $y=5x^2+2x-25$ at the point $(2,-1)$, then $|2a+9b|$ is equal to ________.
Differentiation and its Application
Answer$195$
Solution
Concept. Find the tangent to the parabola at $(2,-1)$; then find where the cubic has the same slope and also lies on that line. Formulas used. (1) Tangent at $(x_1,y_1)$: $y-y_1=y'(x_1)(x-x_1)$Step 1 - Tangent to the parabola. $y'=10x+2=22$ at $x=2$. By (1): $y+1=22(x-2)$, i.e. $y=22x-45$. Step 2 - Same slope on the cubic. $3a^2-2a+1=22$ gives $3a^2-2a-21=0$, so $a=3$ or $a=-\frac73$. Step 3 - On the line. $a=3$: $b=27-9+3=21$ and $22\cdot3-45=21$. Yes. $a=-\frac73$: $b=-\frac{553}{27}$, but the line gives $-\frac{289}{3}$. No. Step 4 - Value. $|2a+9b|=|6+189|=195$. Answer: 195