Conic Sections

Grade 11 · Math · Study Guide

An exploration of parabolas, ellipses, hyperbolas, and circles defined as cross-sections of a double cone.

Conic sections are curves formed by the intersection of a plane with a double cone. The four types are circles, parabolas, ellipses, and hyperbolas. Each has a standard equation and unique geometric properties.

A circle with center (h, k) and radius r: (x-h)² + (y-k)² = r². A parabola with vertex at origin opening upward: x² = 4py, where the focus is at (0, p) and directrix is y = -p. An ellipse centered at (h, k) with semi-major axis a along x: (x-h)²/a² + (y-k)²/b² = 1, where c² = a² - b² and c is the distance from center to each focus. A hyperbola centered at (h, k) with transverse axis along x: (x-h)²/a² - (y-k)²/b² = 1, where c² = a² + b² and the asymptotes are y - k = ±(b/a)(x - h).

The eccentricity e = c/a characterizes each conic: circle e = 0, ellipse 0 < e < 1, parabola e = 1, hyperbola e > 1. Completing the square is a critical technique for converting the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0 into standard form to identify the conic.

Key terms

Focus
A fixed point used in defining parabolas, ellipses, and hyperbolas; plural foci.
Directrix
A fixed line used in the definition of a parabola; every point on the parabola is equidistant from the focus and directrix.
Eccentricity
A non-negative number e = c/a that measures how much a conic deviates from a circle; e = 0 for circle, e = 1 for parabola.
Semi-major Axis
The longer half-axis of an ellipse; its length is a in the standard form.
Asymptotes of Hyperbola
Lines the hyperbola approaches but never touches; for (x²/a²) - (y²/b²) = 1, they are y = ±(b/a)x.
Completing the Square
Algebraic technique to rewrite quadratic expressions in vertex/standard form for conic identification.

Flashcards

Circle equation
(x-h)² + (y-k)² = r²; center (h,k), radius r
Parabola (vertical axis)
x² = 4py; focus at (0,p), directrix y = -p
Ellipse standard form
(x-h)²/a² + (y-k)²/b² = 1; a > b; foci on major axis
Ellipse: c² =
c² = a² - b²; c = distance from center to focus
Hyperbola standard form (horizontal)
(x-h)²/a² - (y-k)²/b² = 1
Hyperbola: c² =
c² = a² + b²
Hyperbola asymptotes
y - k = ±(b/a)(x - h) for horizontal transverse axis
Eccentricity values
Circle: e=0; Ellipse: 0<e<1; Parabola: e=1; Hyperbola: e>1

Questions

  1. What is the equation of a parabola with vertex at the origin and focus at (0, -3)?

    1. x² = 12y
    2. x² = -12y
    3. y² = 12x
    4. y² = -12x
    Show answer

    Answer: x² = -12y

    Focus at (0, -3) means p = -3. Parabola equation: x² = 4py = 4(-3)y = -12y.

  2. For the ellipse x²/25 + y²/9 = 1, the distance from the center to each focus is c = ___.

    Show answer

    Answer: 4

    a² = 25, b² = 9, so c² = a² - b² = 25 - 9 = 16, giving c = 4.

  3. A hyperbola with equation x²/16 - y²/9 = 1 has asymptotes y = ±(3/4)x.

    Show answer

    Answer: True

    For x²/a² - y²/b² = 1 with a=4 and b=3, asymptotes are y = ±(b/a)x = ±(3/4)x.

  4. Which conic is represented by x² + y² - 4x + 6y - 3 = 0?

    1. Parabola
    2. Ellipse
    3. Circle
    4. Hyperbola
    Show answer

    Answer: Circle

    Complete the square: (x-2)² + (y+3)² = 16, a circle with center (2, -3) and radius 4.

  5. The eccentricity of a circle is ___.

    Show answer

    Answer: 0

    For a circle, c = 0 (both foci coincide at center), so e = c/a = 0.

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