Let S be the set of all a N such that the area of the triangle formed by the tangent at th…

Question

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 $16$ unit2, then $\sum _{a\in S}a$ is equal to :

Answer: 146

Accepted Answer

As $P(b,c)$ lies on parabola so ${c}^{2}=2ab---(1)$

Now equation of tangent to parabola ${y}^{2}=2ax$ in point form is

${yy}_{1}=2a\frac{(x+{x}_{1})}{2},$

Here, $({x}_{1},{y}_{1})=(b,c)$

$\Rightarrow yc=a(x+b)$

For point $B$, put $y=0$, now $x=-b$

So, area of $	rianglePBA,\frac{1}{2}	imes AB	imes AP=16$

$\begin{gathered} \Rightarrow \frac{1}{2}	imes 2b	imes c=16 \ \Rightarrow bc=16 \end{gathered}$

As $b$ and $c$ are natural number so possible values of $(b,c)$ are $(1,16),(2,8),(4,4),(8,2)$ and $(16,1)$

Now from equation (1) $a=\frac{{c}^{2}}{2b}$ and $a\in N$, so values of $(b,c)$ are $(1,16),(2,8)$ and $(4,4)$ now values of are 128,16 and 2 .

Hence sum of values of $a$ is 146 .

Solutions · 1