Question

: What is the maximum number of intersection points a parabola and a circle could have?

A: 2
B: 3
C: 4
D: 1

: What is the maximum number of possible solutions for the system shown below?

X^2 – 4y^2 = 64
x^2 + y^2 = 36

A: 4
B: 2
C: 3
D: 1

Answer 1

Answer on Question #57345 – Math – Analytic Geometry

Question

1. What is the maximum number of intersection points a parabola and a circle could have?

A: 2

B: 3

C: 4

D: 1

Solution

The canonical equation of the parabola is $y^{2} = 2px$, the canonical equation of the circle is $x^{2} + y^{2} = a^{2}$.

The intersection points are determined by the solutions of the following system

$$\left\{ \begin{array}{l} y^{2} = 2px \\ x^{2} + y^{2} = a^{2} \end{array} \right. \Rightarrow x = y^{2} / 2p$$ $$\frac{y^{4}}{4p^{4}} + y^{2} = a^{2} \Rightarrow$$

$y^{4} + 4p^{4}y^{2} = 4p^{4}a^{2}$ is equation of the fourth degree, hence the maximum number of intersection points is 4.

Answer: C: 4.

Question

2. What is the maximum number of possible solutions for the system shown below?

$$x^{2} - 4y^{2} = 64$$ $$x^{2} + y^{2} = 36$$

A: 4

B: 2

C: 3

D: 1

Solution

$$\left\{ \begin{array}{l} x^{2} - 4y^{2} = 64 \\ x^{2} + y^{2} = 36 \end{array} \right. \Rightarrow 5y^{2} = -28 \Rightarrow \emptyset.$$

There are no points of intersection, but this type of equations, where we have intersection between hyperbola and circle, we can obtain at most 4 points of intersection.

Answer: A: 4.

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