Conic Sections — Test

Question 17 • Geometry

Which graph represents the hyperbola the fraction with numerator x squared and denominator 3 squared minus the fraction with numerator y squared and denominator 8 squared is equal to 1 ?

Answer

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (negative 8 comma 0) and (8 comma 0). The branches of the hyperbola open left and right. A dashed box is shown around the points (negative 8 comma 0), (0 comma 3), (8 comma 0), and (0 comma negative 3).

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (negative 3 comma 0) and (3 comma 0). The branches of the hyperbola open left and right. A dashed box is shown around the points (negative 3 comma 0), (0 comma 8), (3 comma 0), and (0 comma negative 8).

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (0 comma negative 8) and (0 comma 8). The branches of the hyperbola open up and down. A dashed box is shown around the points (negative 3 comma 0), (0 comma 8), (3 comma 0), and (0 comma negative 8).

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (0 comma negative 3) and (0 comma 3). The branches of the hyperbola open up and down. A dashed box is shown around the points (negative 8 comma 0), (0 comma 3), (8 comma 0), and (0 comma negative 3).

Explanation

The equation is x²/3² − y²/8² = 1, which is in the form x²/a² − y²/b² = 1. Since the x-term is positive, the hyperbola opens left and right, and its transverse-axis vertices are (−a, 0) and (a, 0). Here a = 3, so the vertices are (−3, 0) and (3, 0); choice D shows this, with the dashed box extending 8 units up and down.

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