Let S be the set of all $a \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P(b,c)$, $b,c \in \mathbb{N}$, 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 ___.
Step-by-Step Solution
Key Concept: Tangent at $(b,c)$ on $y^2=2ax$ meets x-axis at $(-b,0)$; triangle base = $2b$, height = $c$, area = $bc = 16$; use $c^2 = 2ab$ to find valid natural number solutions
Area $= \frac{1}{2} \times 2b \times c = bc = 16$. Natural number pairs $(b,c)$ with $bc=16$: $(1,16),(2,8),(4,4),(8,2),(16,1)$. Since $c^2=2ab \Rightarrow a = c^2/(2b)$: for $(1,16): a=128$; $(2,8): a=16$; $(4,4): a=2$; $(8,2): a=1/4$ (not $\in \mathbb{N}$); $(16,1): a=1/32$ (not $\in \mathbb{N}$). So $S = \{128,16,2\}$, $\sum a = 146$. Answer: 146
Correct Answer: 146