The standard equation of an ellipse centered at the origin is ___ + ___ = 1.

X²/a² + y²/b² = 1

True or False: An ellipse has only one focus.

False: An ellipse has two foci.

Fill in the blank: The major axis of an ellipse has a length of ___, while the minor axis has a length of ___.

2a, 2b

What is the relationship between the lengths of the axes of an ellipse?

In an ellipse, a > b, where 'a' is the semi-major axis and 'b' is the semi-minor axis.

Fill in the blank: The eccentricity 'e' of an ellipse is defined as e = ___, where ___ is the distance between the foci and 2a is the distance between the vertices.

√(1 - (b²/a²)), 2ae

True or False: The sum of the distances from any point on the ellipse to the two foci is constant.

True.

In parametric form, the coordinates of a point on the ellipse can be expressed as (___, ___).

(a cos θ, b sin θ)

What defines the directrix of an ellipse, and how is it related to the foci?

The directrix is a line such that the distance from any point on the ellipse to a focus divided by its distance to the directrix is equal to the eccentricity 'e'.

Fill in the blank: The lengths of the latus rectum of an ellipse are given by ___ and are equal to ___.

2b²/a, 2b²/a

What is the geometric significance of the center of an ellipse?

The center of an ellipse is the point where all chords passing through it are bisected.

True or False: The distance between the directrices of an ellipse is equal to 2a.

False: The distance between the directrices is equal to 2a/e.

What is the condition for a line represented by y = mx + c to be tangent to an ellipse?

The condition is that the discriminant of the quadratic equation formed must be zero.

Fill in the blank: The coordinates of the foci of a horizontal ellipse are (___, 0) and (___, 0).

(ae, 0), (-ae, 0)

Explain the significance of the eccentricity of an ellipse.

Eccentricity indicates the degree of elongation of the ellipse; it varies between 0 (circle) and 1 (parabola).

What is the relationship between the coordinates of points on the ellipse and their distances to the foci?

For any point P(x, y) on the ellipse, the sum of the distances to the two foci is constant and equals 2a.