If length of subnormal to the curve $y = x^{2} - 3 x + 2$ at $x = 3$ is equal to the length of subtangent to the curve $y = x^{2} + 2 x + a,$ then the number of non-negative integral value(s) of $a$ is

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Solution

$y = x^{2} - 3 x + 2$
At $x = 3, y = 2$ and $\frac{d y}{d x} = 2 x - 3 = 3$
Now, length of subnormal at $(3, 2)$ is $∣ ∣ ∣ y \frac{d y}{d x} ∣ ∣ ∣ = 6$

$y = x^{2} + 2 x + a$
$\frac{d y}{d x} = 2 x + 2$
So, length of subtangent $= ∣ ∣ ∣ y \cdot \frac{d x}{d y} ∣ ∣ ∣ = 6$
$\Rightarrow \frac{x^{2} + 2 x + a}{2 x + 2} = 6 \Rightarrow x^{2} - 10 x + (a - 12) = 0$
Since there should exist real value of $x$ ,
$D \geq 0$
$\Rightarrow 100 - 4 (a - 12) \geq 0 \Rightarrow a \leq 37$

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