An Interesting Conic Problem!

Geometry Level pending

The equation of a conic is,

(x^2) +116x + 4(y^2) + 2y + 4xy + 259 = 0. Find out what kind of conic it is. Then, Let, the ways we can permutate the letters of the name of This conic is X. Let, the directix of this conic is in the form ax+by+c=0. Let, the focus of This conic is ( p,q ).

Find out the value of ( X+(a)^2+(b)^2+(c)^2+(p)^2+(q)^2 )-172


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