JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Application of Derivatives

Key concept: Critical points and tangent-normal to parabola.

Step 1: Find $a$ from the decreasing/increasing condition.

So $a=\frac{1}{2}$, giving parabola $y^2=2x$.

Step 2: Find point P.
Let $P(x_1,y_1)$ on $y^2=2x$. Tangent equation: $y{y}_1=x+x_1$

Passes through $(4,3)$:


Since $y_1^2=2x_1$:


Check: tangent at $(2,2)$ passes through $(-2,0)$ (rejected).
Tangent at $(8,4)$ does not pass through $(-2,0)$ (accepted).

So $P(8,4)$.

Step 3: Normal at P has slope $-1/m=-4$ where $m=1/y_1=1/4$.



Therefore $\alpha =9,\beta =36$, and $\alpha +\beta =45$.