Ellipse and Hyperbola Flashcard 10.2 and 10.3

Front

What is the standard form of a hyperbola?

Back

The standard form of a hyperbola with a horizontal transverse axis is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) and for a vertical transverse axis is \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \).

Front

How do you find the eccentricity of an ellipse?

Back

The eccentricity \( e \) of an ellipse is calculated using the formula \( e = \frac{c}{a} \), where \( c \) is the distance from the center to a focus and \( a \) is the distance from the center to a vertex.

Front

What are the foci of an ellipse given in standard form?

Back

For an ellipse in standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the foci are located at \( (h \pm c, k) \) if \( a > b \) and at \( (h, k \pm c) \) if \( b > a \), where \( c = \sqrt{a^2 - b^2} \).

Front

What is the relationship between the axes of a hyperbola and its asymptotes?

Back

The asymptotes of a hyperbola are lines that the hyperbola approaches as it extends to infinity. For a hyperbola in standard form, the equations of the asymptotes can be derived from the center and the slopes determined by \( \frac{b}{a} \).

Front

What is the definition of a hyperbola?

Back

A hyperbola is a type of conic section that is formed by the intersection of a plane and a double cone, characterized by two separate curves called branches. It can be defined as the set of all points where the absolute difference of the distances to two fixed points (foci) is constant.

Front

How do you determine the equations of the asymptotes for a hyperbola?

Back

For a hyperbola in standard form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), the equations of the asymptotes are given by \( y - k = \pm \frac{b}{a}(x - h) \).

Front

What is the significance of the transverse axis in a hyperbola?

Back

The transverse axis of a hyperbola is the line segment that connects the two vertices of the hyperbola. It is the axis along which the hyperbola opens and is crucial for determining the hyperbola's orientation.

Front

What is the standard form of an ellipse?

Back

The standard form of an ellipse is given by \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \( (h,k) \) is the center, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis.

Front

How do you convert a general conic equation to standard form?

Back

To convert a general conic equation to standard form, you typically complete the square for both the x and y terms, rearranging the equation to isolate the conic section on one side.

Front

What is the formula for the distance between the foci of a hyperbola?

Back

The distance between the foci of a hyperbola is given by \( 2c \), where \( c = \sqrt{a^2 + b^2} \), with \( a \) and \( b \) being the distances from the center to the vertices and co-vertices, respectively.