Trivia Quiz On Hyperbolas 10.4

A hyperbola is a conic section that has asymptotes. Asymptotes are lines that the hyperbola approaches but never intersects. They are used to define the shape and orientation of the hyperbola. The presence of asymptotes distinguishes a hyperbola from other conic sections like ellipses and parabolas. A hyperbola also has a major axis, which is the line segment connecting the vertices, and two foci. However, the number of foci is not specified in the question, so it is not included in the explanation.

The answer (-3,2) indicates that the center of the given object, which is not specified in the question, is located at the coordinates (-3,2).

The length of the major axis refers to the longest diameter of an ellipse. In this case, the given options represent different lengths for the major axis. Among these options, the correct answer is 6, as it represents the length of the major axis.

The equation x^2 / 4 - y^2 / 19 = 1 represents a hyperbola. The major axis of a hyperbola is the longer axis, which is determined by the term with the larger coefficient. In this case, the term with the larger coefficient is x^2 / 4, so the major axis length is 2.

The given answer states that the vertices are (0,2)(-6,2) and (0,2)(-6,2). This means that there are two sets of coordinates given, (0,2) and (-6,2), and both sets are repeated.

The equation y^2 / 9 - x^2 / 16 = 1 represents a hyperbola with its center at the origin (0,0). The foci of a hyperbola are located along the transverse axis, which is the y-axis in this case. The distance from the center to each focus is given by c, where c^2 = a^2 + b^2, and a and b are the lengths of the conjugate and transverse axes, respectively. In this equation, a = 3 and b = 4. Plugging these values into the equation, we get c^2 = 3^2 + 4^2 = 9 + 16 = 25. Therefore, c = 5. Since the foci are located along the y-axis, the coordinates of the foci are (0,5) and (0,-5).