Question

question 7 of 10

the foci of an ellipse can be outside the ellipse.

a. true

b. false

Explanation:

Analyze the definition of an ellipse and its foci

An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. For any point \(P\) on the ellipse and foci \(F_1, F_2\), the relation is \(PF_1 + PF_2 = 2a\), where \(2a\) is the length of the major axis.

Determine the position of the foci relative to the boundary

The distance between the two foci is \(2c\). For a non-degenerate ellipse, the inequality \(2c < 2a\) must hold, which simplifies to \(c < a\). Since the distance from the center to each focus \(c\) is strictly less than the semi-major axis \(a\), the foci must always lie inside the boundary of the ellipse.

Evaluate the statement

The statement "The foci of an ellipse can be outside the ellipse" is false because the foci must always lie on the major axis strictly between the vertices, which places them inside the interior region of the ellipse.

Answer:

• A. True
• B. False (Correct answer)