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 $\displaystyle\sum_{a \in S} a$ is equal to _____

Answer 1

As $P ( b , c )$ lies on parabola so $c ^2=2 ab$....(1)
Now equation of tangent to parabola $y^2=2 ax$ in point
$ \text { form is } yy _1=2 a \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 $ \triangle PBA , \frac{1}{2} \times AB \times AP =16$
$ \Rightarrow \frac{1}{2} \times 2 b \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) $a =\frac{ c ^2}{2 b }$ 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 $146$ .