Parabola Question 6
Question 6 - 31 January - Shift 2
Let $S$ be the set of all $a \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P(b, c), b, c \in \mathbb{N}$, 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 ________
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Answer: 146
Solution:
Formula: Equation of the Tangent at any Point, Area of Triangle
As $P(b, c)$ lies on parabola so $c^{2}=2 ab \cdots (1)$
Now equation of tangent to parabola $y^{2}=2 a x$ in point
form is $yy_1=2 a \frac{(x+x_1)}{2},(x_1, y_1)=(b, c)$
$\Rightarrow yc=a(x+b)$
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 $a \in \mathbb{N}$, so values 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 .
