What is the geometric locus definition of an ellipse?
An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (called foci) is constant: \(d_1 + d_2 = 2a\).
What is the standard equation of an ellipse centered at \((h, k)\) with a horizontal major axis?
$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad (a > b > 0)$$
What is the standard equation of an ellipse centered at \((h, k)\) with a vertical major axis?
$$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \quad (a > b > 0)$$
In the standard equation of an ellipse, how do you determine whether the major axis is horizontal or vertical?
The major axis aligns with the variable whose fraction has the larger denominator (\(a^2\)). If \(a^2\) is beneath the \(x\)-term, it is horizontal; if beneath the \(y\)-term, it is vertical.
What equation relates the semi-major axis \(a\), semi-minor axis \(b\), and focal distance \(c\) in an ellipse?
$$c^2 = a^2 - b^2$$
For an ellipse centered at \((h, k)\) with a horizontal major axis, what are the coordinates of the vertices?
$$(h \pm a, k)$$
For an ellipse centered at \((h, k)\) with a vertical major axis, what are the coordinates of the vertices?
$$(h, k \pm a)$$
For an ellipse centered at \((h, k)\) with a horizontal major axis, what are the coordinates of the foci?
$$(h \pm c, k)$$
For an ellipse centered at \((h, k)\) with a vertical major axis, what are the coordinates of the foci?
$$(h, k \pm c)$$
For an ellipse centered at \((h, k)\) with a horizontal major axis, what are the coordinates of the co-vertices?
$$(h, k \pm b)$$
In an ellipse with semi-major axis \(a\) and semi-minor axis \(b\), what are the total lengths of the major and minor axes?
The length of the major axis is \(2a\), and the length of the minor axis is \(2b\).
What is the formula for the eccentricity \(e\) of an ellipse?
$$e = \frac{c}{a}$$
What range of values can the eccentricity \(e\) of an ellipse take?
$$0 \le e < 1$$
A circle has \(e = 0\), whereas values closer to \(1\) produce a more elongated, flatter ellipse.
What algebraic technique transforms a general equation \(Ax^2 + Cy^2 + Dx + Ey + F = 0\) into the standard form of an ellipse?
Completing the square on both the \(x\) and \(y\) variable groupings, followed by dividing the entire equation by the resulting constant term on the opposite side.
Find the focal distance \(c\) for the ellipse: $$\frac{x^2}{25} + \frac{y^2}{9} = 1$$
