Analytic Geometry (Conic Sections) Flashcards

The equation of an ellipse is x²/25 + y²/16 = 1. What is the distance between its two foci?

Answer: 6

For x²/a² + y²/b² = 1 with a > b, c² = a² - b² = 25 - 16 = 9, so c = 3. The distance between foci is 2c = 6.

A parabola has the equation y² = 8x. What are the coordinates of its focus?

Answer: (2, 0)

The standard form y² = 4px gives focus at (p, 0). Here 4p = 8 so p = 2, giving focus (2, 0).

Which of the following is the equation of a hyperbola with real axis along the y-axis?

Answer: y²/9 - x²/4 = 1

A hyperbola with real axis along the y-axis has the form y²/a² - x²/b² = 1. Option A matches this form.

If the eccentricity of an ellipse is 1/2, and one semi-axis is a = 4, what is the length of the other semi-axis b?

Answer: 2√3

Eccentricity e = c/a = 1/2, so c = 2. Then b² = a² - c² = 16 - 4 = 12, giving b = 2√3.

A circle passes through the origin and has center at (3, 4). What is the equation of this circle?

Answer: (x-3)² + (y-4)² = 25

The radius is the distance from center (3,4) to origin (0,0): r = √(9+16) = 5. Equation: (x-3)² + (y-4)² = 25.

The asymptotes of the hyperbola x²/4 - y²/9 = 1 are:

Answer: y = ±(3/2)x

For x²/a² - y²/b² = 1, asymptotes are y = ±(b/a)x. Here a = 2, b = 3, so y = ±(3/2)x.