Ellipse Review (11/15/24)

Front

What is the standard form of the equation of an ellipse centered at the origin?

Back

The standard form is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

Front

What does it mean for an ellipse to be elongated horizontally?

Back

It means that the length of the semi-major axis (\( a \)) is greater than the length of the semi-minor axis (\( b \)), and the standard form of the equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with \( a^2 \) in the denominator of the \( x^2 \) term.

Front

What is the definition of an ellipse?

Back

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant.

Front

How do you find the endpoints of the major axis of an ellipse?

Back

The endpoints of the major axis are located at (\( h \pm a \), \( k \)) for a horizontally oriented ellipse and (\( h \), \( k \pm a \)) for a vertically oriented ellipse, where (\( h, k \)) is the center of the ellipse.

Front

What are the coordinates of the foci of an ellipse?

Back

The foci are located at (\( h \pm c \), \( k \)) for a horizontally oriented ellipse and (\( h \), \( k \pm c \)) for a vertically oriented ellipse, where \( c = \sqrt{a^2 - b^2} \).

Front

What is the relationship between the semi-major axis, semi-minor axis, and the distance to the foci in an ellipse?

Back

In an ellipse, the relationship is given by \( c^2 = a^2 - b^2 \), where \( c \) is the distance from the center to each focus, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis.

Front

What happens to the equation of an ellipse when it is translated?

Back

When an ellipse is translated, the center (h, k) changes, and the standard form of the equation becomes \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).

Front

How do you determine the length of the minor axis of an ellipse?

Back

The length of the minor axis is \( 2b \), where \( b \) is the semi-minor axis.

Front

What is the significance of the values \( a \) and \( b \) in the equation of an ellipse?

Back

The value \( a \) represents the length of the semi-major axis, while \( b \) represents the length of the semi-minor axis.

Front

What is the general form of the equation of an ellipse?

Back

The general form is \( Ax^2 + By^2 + Cx + Dy + E = 0 \) where A and B are positive constants.