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

The correct answer is 146.

As P(b,c) lies on parabola so c2=2ab....(1)
Now equation of tangent to parabola y2=2ax in point
form is yy1​=2a2(x+x1​)​,(x1​,y1​)=(b,c)
⇒yc=a(x+b)
For point B, put y=0, now x=−b
So, area of △PBA,21​×AB×AP=16
⇒21​×2b×c=16
⇒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=2bc2​ and a∈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 .