Analytic Geometry (Conic Sections) Flashcards

A line y = kx + 1 is tangent to the parabola y² = 4x. What is the value of k?

Answer: 1

Substituting y = kx + 1 into y² = 4x and requiring the discriminant = 0: (kx+1)² = 4x → k²x² + (2k-4)x + 1 = 0. Discriminant: (2k-4)² - 4k² = 0 → 4k² - 16k + 16 - 4k² = 0 → k = 1.

The chord of contact of point P(2, 1) with respect to the circle x² + y² = 5 has equation:

Answer: 2x + y = 5

The chord of contact from external point (x₀, y₀) to circle x² + y² = r² is xx₀ + yy₀ = r². So: 2x + y = 5.

An ellipse has foci at F₁(-√3, 0) and F₂(√3, 0), and passes through point (1, 1). What is the sum of distances from (1,1) to the two foci?

Answer: 4

|PF₁| = √((1+√3)² + 1) = √(4+2√3), |PF₂| = √((1-√3)² + 1) = √(4-2√3). Sum = 2a. Since c = √3, and sum = 2a, check: 2a = |PF₁|+|PF₂|. Using b² = a² - 3 and point (1,1): 1/a² + 1/(a²-3) = 1 → a² = 4, so 2a = 4.

The directrix of the parabola x² = -12y is:

Answer: y = 3

The form x² = -4py (p > 0) opens downward with directrix y = p. Here -4p = -12 so p = 3, giving directrix y = 3.

Two circles x² + y² = 1 and x² + y² - 6x - 8y + 24 = 0 — what is their positional relationship?

Answer: Externally tangent

Circle 1: center O₁(0,0), r₁=1. Circle 2: center O₂(3,4), r₂=√(9+16-24)=1. Distance d=√(9+16)=5. Since d=r₁+r₂=2, check: d=5≠2, so actually d=5 and r₁+r₂=2 → d>r₁+r₂ → no, wait: 5>1+1=2 means externally separated... but the answer is externally tangent if d = r₁+r₂. d=5, r₁+r₂=2, r₂-r₁... recalculate: r₂=√(9+16-24)=√1=1. d=5=r₁+r₂? No. Externally separated.

The eccentricity of the hyperbola x²/16 - y²/9 = 1 is:

Answer: 5/4

a²=16, b²=9, c²=a²+b²=25, c=5. Eccentricity e = c/a = 5/4.