Which of the following circles has the greatest number of points of intersection with the parabola $x^{2} = y + 4$

Question

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

Answer 1

You should hopefully have noticed that all the given equations are circles centred at $(0,0)$ with radius $A=1$, $B=\sqrt2$, $C=3$, $D=4$, $E=5$.

We can quite easily plot the circles on a graph, and the initial parabola shouldn't provide many problems either:

(graph from WolframAlpha)

From there, we can very quickly see which of the circles intersects with the parabola the most times

Answer 2

Hint: I think you may try plotting the graph. Not sure if the smartest way but you will clearly notice the vertex of parabola at (0,-4). Now try plotting the equations and they may do the job. (Constructive suggestion to the answer are welcome)

Answer 3

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:

$$ (\dfrac{\sqrt15}{2}<R < 4 )$$