The parabola
is tangent to the graph of
at two distinct points. Given that the
area enclosed by these two curves is
, where
and
are coprime positive integers, find the value of
.
Remark : try the case for here
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for the minimum area g(x) should be symmeyric about y axis as parabola so b= 0 =a now at point of inter section x^2=x^4+cx^2+1 as they touch this equation have pair of coinside root so c=-1 eqution reduce to x^4-2x^2+1=0 ie (x^2-1)^2=0 x=1,1,-1,-1 and requred area $x^4-x^2+1-x^2dx from x=-1 to x=1 = 16/15 p=16 ,q=15 so p+q= 15+16=31