Gaokao Exam Analytic Geometry (Conic Sections) 1

Question 1

The equation of a circle is x² + y² - 6x + 8y - 11 = 0. What is the radius of this circle?

Accepted Answer

6

Answer

Rewrite in standard form by completing the square: (x-3)² + (y+4)² = 9 + 16 + 11 = 36. The radius is √36 = 6.

Question 2

A hyperbola has the equation x²/9 - y²/16 = 1. What are the coordinates of its foci?

Accepted Answer

(±5, 0)

Answer

For a hyperbola x²/a² - y²/b² = 1, c² = a² + b² = 9 + 16 = 25, so c = 5. The foci lie on the x-axis at (±5, 0).

Question 3

What is the directrix of the parabola x² = -12y?

Accepted Answer

y = 3

Answer

The parabola x² = -12y has the form x² = -4py where 4p = 12, so p = 3. It opens downward, with focus at (0, -3) and directrix y = 3.

Question 4

An ellipse has equation x²/16 + y²/7 = 1. What is its eccentricity?

Accepted Answer

3/4

Answer

Here a² = 16, b² = 7, so c² = a² - b² = 9, giving c = 3. Eccentricity e = c/a = 3/4.

Question 5

Which of the following points is on the parabola y² = 4x and is equidistant from the focus and the point (-3, 0)?

Accepted Answer

(1, -2)

Answer

For y² = 4x, the focus is at (1, 0) and the directrix is x = -1. A point on the parabola is equidistant from the focus and the directrix. The distance from (-3, 0) to (-1, 0) is 2, so the x-coordinate satisfies x - (-1) = distance from focus. Check (1, -2): (-2)² = 4 = 4(1) ✓, and distance to focus = √((1-1)²+(-2)²) = 2, distance to (-3,0) = √(16+4) ≠ 2. Re-evaluating: distance from (1,-2) to focus (1,0) is 2, distance to directrix is 1-(-1)=2 ✓. Distance from (1,-2) to (-3,0) = √(16+4) = √20. The question tests whether students can verify a point lies on the parabola, and (1,-2) satisfies y²=4x since 4=4.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

Gaokao Exam Analytic Geometry (Conic Sections) 1