<s>[INST] Which of the following circles has the greatest number of points of intersection with the parabola $x^{2} = y + 4$ Which of the following circles has the greatest number of points of intersection with the parabola $x^2 = y + 4$?
(A) $x^2 + y^2=1$
(B) $x^2 + y^2=2$
(C) $x^2 + y^2=9$
(D) $x^2 + y^2=16$
(E) $x^2 + y^2=25$ 
I tried  to solve every equation by substituting the value of $x^2$ in it, but is this the smartest way to solve this question?
The question is supposed to be answered in no more than $2.5$ minutes    
 [/INST] Eliminate $x$ between them leading to quadratic equation
$$ x^2+y^2= R^2,\, x^2= y+4 ,\, \rightarrow y^2+ y + 4- R^2 =0 $$
If its discriminant is $>0$ then real roots occur when
$$ R>\frac{\sqrt15} {2} \approx  1.9365$$
Repeated roots occur when $ R = \dfrac{\sqrt15}{2}  $ 
But graphically repeated roots occur also for 
$$ R= 4 $$
EDIT 1:
For a comprehensive  algebraic ( analytical geometrical)  procedure (above is incomplete),
we next consider symmetry along y- axis, $x=0$
$$ y=-4, x=0, \rightarrow \, R=4 $$ 
which supplies another tangent point.
So the complete interval to be considered for maximum real roots cutting situation is:
$$ (\dfrac{\sqrt15}{2}<R  < 4 )$$

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