JEE Main 2023 (Online) 31st January Evening ShiftQuestion 1
Let $S$ be the set of all $a\in N$ such that the area of the triangle formed by the tangent at the point $P(b,c)$, $b,c\in N$, on the parabola $y^{2}=2ax$ and the lines $x=b$, $y=0$ is $16$ unit$^{2}$, then $\sum_{a\in S}a$ is equal to
Parabola
Answer$146$
Solution
Tangent at $(b,c)$: $cy=a(x+b)$ meets $y=0$ at $x=-b$, so the triangle has vertices $(-b,0),(b,0),(b,c)$ and area $bc=16$. With $c^2=2ab$, $a=\frac{c^2}{2b}$: $(b,c)=(1,16),(2,8),(4,4)$ give $a=128,16,2$. Sum $=146$.