What is the geometric locus definition of a hyperbola?
A hyperbola is the set of all points in a plane such that the absolute difference of the distances from two fixed points (foci) is constant: \(|d_1 - d_2| = 2a\).
What is the standard equation of a hyperbola centered at \((h, k)\) with a horizontal transverse axis?
How do you determine whether a hyperbola opens horizontally or vertically from its standard equation?
Look at the sign of the squared terms. If the \(x\)-term is positive, the hyperbola opens horizontally. If the \(y\)-term is positive, it opens vertically.
In a hyperbola equation, must \(a\) always be greater than \(b\)?
No. Unlike an ellipse, \(a\) does not need to exceed \(b\). The parameter \(a^2\) is always the denominator of the positive fraction.
What equation relates the parameters \(a\), \(b\), and \(c\) in any hyperbola?
$$c^2 = a^2 + b^2$$
What are the lengths of the transverse axis and the conjugate axis in a hyperbola?
What are the coordinates of the vertices for a horizontal hyperbola centered at \((h, k)\)?
$$(h \pm a, k)$$
What are the coordinates of the vertices for a vertical hyperbola centered at \((h, k)\)?
$$(h, k \pm a)$$
What are the coordinates of the foci for a horizontal hyperbola centered at \((h, k)\)?
$$(h \pm c, k)$$
What are the coordinates of the foci for a vertical hyperbola centered at \((h, k)\)?
$$(h, k \pm c)$$
What is the formula and range for the eccentricity \(e\) of a hyperbola?
$$e = \frac{c}{a} > 1$$
Because \(c > a\) in all hyperbolas, the eccentricity is always strictly greater than \(1\).
What is the purpose of drawing the central reference box when sketching a hyperbola?
The box has dimensions \(2a\) by \(2b\) centered at \((h, k)\). Drawing lines through its diagonals provides the asymptotes that guide the hyperbola's branches.
Find the focal distance \(c\) for the hyperbola: $$\frac{y^2}{16} - \frac{x^2}{9} = 1$$
