Conics: Circles, Ellipses, and Hyperbolas RW

Front

What is the standard form of the equation of a hyperbola with vertices at (±a, 0) and foci at (±c, 0)?

Back

The standard form is x²/a² - y²/b² = 1, where c² = a² + b².

Front

Define the term 'asymptotes' in relation to hyperbolas.

Back

Asymptotes are lines that the hyperbola approaches but never intersects. They provide a guide for the shape of the hyperbola.

Front

What is the difference between a vertical and horizontal hyperbola?

Back

A vertical hyperbola opens upwards and downwards (y-axis), while a horizontal hyperbola opens left and right (x-axis).

Front

Given the vertices (±5, 0) and one focus at (6, 0), determine the value of 'a' for the hyperbola.

Back

a = 5.

Front

What is the relationship between the foci, vertices, and the value of 'c' in a hyperbola?

Back

c is the distance from the center to each focus, and c² = a² + b².

Front

How do you find the equation of a hyperbola given its foci and vertices?

Back

Identify 'a' from the vertices, calculate 'c' from the foci, then use c² = a² + b² to find 'b', and write the equation in standard form.

Front

What are the coordinates of the foci for a hyperbola with a horizontal transverse axis?

Back

The foci are located at (±c, 0) where c is calculated from c² = a² + b².

Front

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

Back

The general form is (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center.

Front

How do you determine the major and minor axes of an ellipse?

Back

The major axis is the longest diameter, and the minor axis is the shortest diameter, determined by the values of 'a' and 'b' in the ellipse equation.

Front

What is the significance of the center in conic sections?

Back

The center is the midpoint of the major and minor axes and serves as the reference point for defining the conic's properties.