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Pre-Calculus

Conic Sections: Parabola, Ellipse, and Hyperbola

Pre-Calculus

  • ✓By the end of this lesson students will be able to define and identify the key features of parabolas, ellipses, and hyperbolas.
  • ✓By the end of this lesson students will be able to write the standard form equations for parabolas, ellipses, and hyperbolas given specific characteristics.
  • ✓By the end of this lesson students will be able to graph parabolas, ellipses, and hyperbolas by identifying their centers, vertices, foci, and asymptotes (for hyperbolas).
  • ✓By the end of this lesson students will be able to convert general form equations of conic sections into standard form by completing the square.
  • ✓By the end of this lesson students will be able to classify a conic section from its general form equation.

Key concepts

Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. The vertex is the midpoint between the focus and the directrix. The axis of symmetry is the line passing through the focus and the vertex, perpendicular to the directrix. The value 'p' represents the directed distance from the vertex to the focus. If p > 0, the parabola opens in the positive direction; if p < 0, it opens in the negative direction.

Standard Forms:\nVertical axis: (x - h)^2 = 4p(y - k)\n Vertex: (h, k)\n Focus: (h, k + p)\n Directrix: y = k - p\nHorizontal axis: (y - k)^2 = 4p(x - h)\n Vertex: (h, k)\n Focus: (h + p, k)\n Directrix: x = h - p
Ellipse

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points, called the foci, is constant. The center of the ellipse is the midpoint of the segment connecting the foci. The major axis is the longer axis of the ellipse, passing through the foci and vertices. The minor axis is the shorter axis, perpendicular to the major axis and passing through the center. 'a' is half the length of the major axis, 'b' is half the length of the minor axis, and 'c' is the distance from the center to each focus. For an ellipse, a > b.

Standard Forms:\nHorizontal major axis: ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1\n Center: (h, k)\n Vertices: (h ± a, k)\n Co-vertices: (h, k ± b)\n Foci: (h ± c, k)\nVertical major axis: ((x - h)^2 / b^2) + ((y - k)^2 / a^2) = 1\n Center: (h, k)\n Vertices: (h, k ± a)\n Co-vertices: (h ± b, k)\n Foci: (h, k ± c)\nRelationship: c^2 = a^2 - b^2
Hyperbola

A hyperbola is the set of all points in a plane such that the absolute difference of the distances from two fixed points, called the foci, is constant. The center of the hyperbola is the midpoint of the segment connecting the foci. The transverse axis is the segment connecting the two vertices, passing through the foci and the center. The conjugate axis is perpendicular to the transverse axis and passes through the center. 'a' is half the length of the transverse axis, 'b' is half the length of the conjugate axis, and 'c' is the distance from the center to each focus. Hyperbolas have two asymptotes that pass through the center and guide the branches of the hyperbola.

Standard Forms:\nHorizontal transverse axis: ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1\n Center: (h, k)\n Vertices: (h ± a, k)\n Foci: (h ± c, k)\n Asymptotes: y - k = ±(b/a)(x - h)\nVertical transverse axis: ((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1\n Center: (h, k)\n Vertices: (h, k ± a)\n Foci: (h, k ± c)\n Asymptotes: y - k = ±(a/b)(x - h)\nRelationship: c^2 = a^2 + b^2
General Form and Classification of Conic Sections

The general form of a conic section is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. In Pre-Calculus, we typically focus on conics where B = 0, meaning the axes are parallel to the coordinate axes. The type of conic can be classified based on the coefficients A and C:\n- If A = 0 or C = 0 (but not both), the conic is a parabola.\n- If A and C have the same sign (AC > 0), the conic is an ellipse (or a circle if A = C).\n- If A and C have opposite signs (AC < 0), the conic is a hyperbola.

Ax^2 + Cy^2 + Dx + Ey + F = 0\n Parabola: AC = 0\n Ellipse: AC > 0\n Hyperbola: AC < 0

Key facts to remember

  • 1A parabola is defined by its focus and directrix.
  • 2An ellipse is defined by the constant sum of distances from two foci.
  • 3A hyperbola is defined by the constant absolute difference of distances from two foci.
  • 4For an ellipse, c^2 = a^2 - b^2, where 'a' is always the larger semi-axis (half the major axis).
  • 5For a hyperbola, c^2 = a^2 + b^2, where 'a' is half the transverse axis (associated with the positive term).
  • 6The general form Ax^2 + Cy^2 + Dx + Ey + F = 0 classifies conics: Parabola if AC=0, Ellipse if AC>0, Hyperbola if AC<0.
  • 7The value 'p' in a parabola's equation is the directed distance from the vertex to the focus.
  • 8Asymptotes are key features for graphing hyperbolas, guiding the branches towards infinity.

Worked examples

Example 1

Find the standard form equation of the parabola with focus (3, 2) and directrix x = -1.

IIdentify the orientation: Since the directrix is a vertical line (x = -1) and the focus is to its right, the parabola opens horizontally to the right. The standard form is (y - k)^2 = 4p(x - h).
IIFind the vertex (h, k): The vertex is halfway between the focus (3, 2) and the directrix x = -1. The y-coordinate of the vertex is the same as the focus, k = 2. The x-coordinate of the vertex is the average of the x-coordinate of the focus and the x-value of the directrix: h = (3 + (-1)) / 2 = 2/2 = 1. So, the vertex is (1, 2).
IIIFind p: The distance from the vertex (1, 2) to the focus (3, 2) is p. p = 3 - 1 = 2. Since the parabola opens to the right, p is positive.
IVSubstitute h, k, and p into the standard form equation: (y - 2)^2 = 4(2)(x - 1).
VSimplify the equation.

Answer

(y - 2)^2 = 8(x - 1)

Always sketch the given information to help visualize the orientation and location of the conic section.

Example 2

Convert the general form equation 9x^2 + 4y^2 - 18x + 16y - 11 = 0 into standard form and identify the conic section.

IClassify the conic: A = 9, C = 4. Since AC = 9 * 4 = 36 > 0, this is an ellipse.
IIGroup x-terms and y-terms, and move the constant to the right side: (9x^2 - 18x) + (4y^2 + 16y) = 11.
IIIFactor out the leading coefficients from the x-terms and y-terms: 9(x^2 - 2x) + 4(y^2 + 4y) = 11.
IVComplete the square for both x and y. For x^2 - 2x, add (-2/2)^2 = 1 inside the parenthesis. For y^2 + 4y, add (4/2)^2 = 4 inside the parenthesis. Remember to balance the equation by adding 9*1 and 4*4 to the right side: 9(x^2 - 2x + 1) + 4(y^2 + 4y + 4) = 11 + 9(1) + 4(4).
VRewrite the expressions in parentheses as squared terms: 9(x - 1)^2 + 4(y + 2)^2 = 11 + 9 + 16.
VISimplify the right side: 9(x - 1)^2 + 4(y + 2)^2 = 36.
VIIDivide both sides by the constant on the right (36) to make the right side equal to 1: (9(x - 1)^2 / 36) + (4(y + 2)^2 / 36) = 36 / 36.
VIIISimplify the fractions.

Answer

((x - 1)^2 / 4) + ((y + 2)^2 / 9) = 1. This is an ellipse with center (1, -2), a vertical major axis (since 9 > 4), a = 3, b = 2.

When completing the square, remember to multiply the added constant by the factored-out coefficient before adding it to the other side of the equation.

Example 3

Find the standard form equation of the hyperbola with vertices at (0, ±6) and foci at (0, ±10).

IIdentify the center: The vertices and foci are symmetric about the origin, so the center (h, k) is (0, 0).
IIIdentify the orientation: Since the vertices and foci lie on the y-axis, the transverse axis is vertical. The standard form is ((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1.
IIIDetermine 'a': The distance from the center (0, 0) to a vertex (0, 6) is a = 6.
IVDetermine 'c': The distance from the center (0, 0) to a focus (0, 10) is c = 10.
VUse the relationship c^2 = a^2 + b^2 to find 'b': 10^2 = 6^2 + b^2. So, 100 = 36 + b^2. This gives b^2 = 100 - 36 = 64, so b = 8.
VISubstitute h, k, a^2, and b^2 into the standard form equation: ((y - 0)^2 / 6^2) - ((x - 0)^2 / 8^2) = 1.
VIISimplify the equation.

Answer

(y^2 / 36) - (x^2 / 64) = 1

For hyperbolas, 'a' is always associated with the positive term and defines the vertices, regardless of whether a > b or b > a.

Common mistakes

  • ✗Confusing the relationship for 'c' between ellipses (c^2 = a^2 - b^2) and hyperbolas (c^2 = a^2 + b^2).
  • ✗Incorrectly identifying 'a' and 'b' in ellipses and hyperbolas, especially which value corresponds to the major/transverse axis.
  • ✗Errors in completing the square, particularly forgetting to multiply the added constant by the factored-out coefficient when balancing the equation.
  • ✗Mixing up the standard forms for horizontal versus vertical orientations, leading to incorrect placement of vertices or foci.
  • ✗Forgetting the 'minus' sign between the squared terms in the hyperbola's standard form equation.

Exam tips

  • ★Always start by classifying the conic section from its general form (if given) to guide your approach.
  • ★Sketch the given information (foci, vertices, directrix) to visualize the conic's orientation and center.
  • ★Memorize the standard form equations and the relationships between a, b, and c for each conic section.
  • ★Be meticulous with algebraic steps, especially when completing the square and simplifying fractions.
  • ★For hyperbolas, remember to find and sketch the asymptotes to accurately draw the graph.

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