Ellipse and Hyperbola Flashcard 10.2 and 10.3

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

How do you determine the values of 'a' and 'b' for an ellipse given its vertices?

Back

The distance between the vertices is \( 2a \). To find \( b \), use the relationship \( c^2 = a^2 - b^2 \) where \( c \) is the distance from the center to the foci.

Front

What is the relationship between 'a', 'b', and 'c' in an ellipse?

Back

The relationship is given by the equation \( c^2 = a^2 - b^2 \), where \( c \) is the distance from the center to each focus.

Front

What is the standard form of the equation of a hyperbola centered at the origin with a horizontal transverse axis?

Back

The standard form is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) where \( a \) is the distance from the center to the vertices along the x-axis.

Front

How do you find the asymptotes of a hyperbola with a horizontal transverse axis?

Back

The equations of the asymptotes are given by \( y = \pm \frac{b}{a} x \) where \( a \) and \( b \) are from the standard form of the hyperbola.

Front

What is the standard form of the equation of a hyperbola centered at the origin with a vertical transverse axis?

Back

The standard form is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) where \( a \) is the distance from the center to the vertices along the y-axis.

Front

How do you determine the values of 'a' and 'b' for a hyperbola given its vertices and foci?

Back

The distance between the vertices is \( 2a \) and the distance between the foci is \( 2c \). Use the relationship \( c^2 = a^2 + b^2 \) to find \( b \).

Front

What is the equation to find 'c' for the focus points on a hyperbola?

Back

The equation is \( c^2 = a^2 + b^2 \), where \( c \) is the distance from the center to each focus.

Front

What are the characteristics of an ellipse?

Back

An ellipse has two foci, a major axis, a minor axis, and the sum of the distances from any point on the ellipse to the two foci is constant.

Front

What are the characteristics of a hyperbola?

Back

A hyperbola has two branches, two foci, and the difference of the distances from any point on the hyperbola to the two foci is constant.