# Modulus+Logs+Circles+Vectors!

Calculus Level 4

Let $$k_1, k_2$$ be any two integers given in the relation $$|(n-a)!-t| + |t-(n-b)!| + | a+b-k_1n-k_2| \leq 0$$ $$\forall n$$ such that $$a < b \leq n$$ and $$a,b,n,t \in N$$.

Let $$P , Q$$ be any two points on the curve

$$y = \log_{\frac{1}{2}} \left({x + \dfrac{k_2}{2}}\right) + \log_2 \sqrt{4x^2 + 4k_2x +(k_1 + k_2)}$$.

Also $$P$$ lies on the circle $$x^2 + y^2 = k_1^3 - 2k_2$$ and $$Q$$ lies inside the given circle such that its abscissa is a non-zero integer.

O is the centre of the circle.

1) Find the minimum possible value of $$\vec{OP} \cdot \vec{OQ}$$.

2) The maximum length of PQ

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