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 $y^2=2$ ax and the lines $x=b,y=0$ is $16$ unit ${}^2$, then $\sum _{a\in S}a$ is equal to _____

Answer 1

Answer: 146

As $P(b,c)$ lies on parabola so $c^2=2ab$....(1)
Now equation of tangent to parabola $y^2=2ax$ in point
$\text{ form is }y{y}_1=2a\frac{\left(x+x_1\right)}{2},\left(x_1,y_1\right)=(b,c)$
$\Rightarrow yc=a(x+b)$
For point $B$, put $y=0$, now $x=-b$
So, area of $△PBA,\frac{1}{2}\times AB\times AP=16$
$\Rightarrow \frac{1}{2}\times 2b\times c=16$
$\Rightarrow bc=16$
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 $146$ .