Let S be the set of all a N such that the area of the triangle formed by the tangent at the point P ( b , c ) , b , c , on the parabola y 2 = 2 a x and the lines x = b , y = 0 is 16 unit 2 , then a S a is equal to _________ . [20

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Solution

Q.
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 $y^{2} = 2 a x$ and the lines $x = b,$ $y = 0$ is 16 ${unit}^{2}$ , then $\sum_{a \in S} a$ is equal to _________ . [2023]

Ans.
(146)

As $P (b, c)$ lies on parabola

So, $c^{2} = 2 a b$ ...(i)

Now, equation of tangent to parabola $y^{2} = 2 a x$ in point form is $y y_{1} = 2 a (\frac{x + x_{1}}{2}), where (x_{1}, y_{1}) \equiv (b, c)$

$\Rightarrow y c = a (x + b)$

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

So, area of $∆ P B A$ , $\frac{1}{2} \times A B \times A P = 16 (Given)$

$\Rightarrow \frac{1}{2} \times 2 b \times c = 16 \Rightarrow b c = 16$

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

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

Hence, sum of values of $a$ is $146$ .

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