Modulus+Logs+Circles+Vectors!

Calculus Level 5

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 length of OP.OQ

2) The maximum length of PQ

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