FREE Gaokao Analytic Geometry (Conic Sections) Questions and Answers

Question 1

An ellipse has foci at F1(-1, 0) and F2(1, 0). The perimeter of a triangle formed by a point P on the ellipse and the two foci is 10. What is the standard equation of the ellipse?

Accepted Answer

x²/16 + y²/15 = 1

Answer

The perimeter of ΔPF1F2 is PF1 + PF2 + |F1F2|. By the definition of an ellipse, PF1 + PF2 = 2a. The distance between the foci, |F1F2|, is 2c. Here, 2c = 2, so c=1. The perimeter is 2a + 2c = 10, so 2a + 2 = 10, which gives 2a = 8 and a = 4. For an ellipse, b² = a² - c² = 4² - 1² = 15. The standard equation is x²/a² + y²/b² = 1.

Question 2

The line y = kx + 1 intersects the parabola y² = 4x at exactly one point. What is the value of k (where k ≠ 0)?

Accepted Answer

k = 1

Answer

Substitute the line equation into the parabola equation: (kx + 1)² = 4x. This expands to k²x² + 2kx + 1 = 4x, or k²x² + (2k - 4)x + 1 = 0. For the line to be tangent (intersect at one point), the discriminant (Δ) of this quadratic equation must be zero. Δ = b² - 4ac = (2k - 4)² - 4(k²)(1) = 0. Expanding this gives 4k² - 16k + 16 - 4k² = 0, which simplifies to -16k + 16 = 0, so k = 1.

Question 3

What are the equations of the asymptotes for the hyperbola x²/9 - y²/16 = 1?

Accepted Answer

y = ±(4/3)x

Answer

For a hyperbola of the form x²/a² - y²/b² = 1, the equations of the asymptotes are y = ±(b/a)x. In this case, a² = 9, so a = 3, and b² = 16, so b = 4. Therefore, the asymptotes are y = ±(4/3)x. This can be rewritten as 3y = ±4x, or 4x ± 3y = 0.

Question 4

The distance between the two foci of an ellipse is 8, and its eccentricity is 4/5. What is the length of the major axis?

Accepted Answer

10

Answer

The distance between the foci is 2c, so 2c = 8, which means c = 4. The eccentricity e is defined as c/a. We are given e = 4/5. So, c/a = 4/a = 4/5. Solving for a gives a = 5. The length of the major axis is 2a, which is 2 * 5 = 10.

Question 5

A point M is on the parabola x² = 8y. If the distance from M to the x-axis is 5, what is the distance from M to the focus F of the parabola?

Accepted Answer

7

Answer

The equation is of the form x² = 2py, so 2p = 8, and p = 4. The focus F is at (0, p/2) = (0, 2), and the directrix is the line y = -p/2 = -2. The distance from M to the x-axis is its y-coordinate, so M has coordinates (x, 5). By the definition of a parabola, the distance from any point on it to the focus is equal to its distance to the directrix. The distance from M(x, 5) to the directrix y = -2 is |5 - (-2)| = 7.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

FREE Gaokao Analytic Geometry (Conic Sections) Questions and Answers