Conic Sections

Outcome 5: Conic Sections · Blueprint Pillar 3 · PGCC MAT-1350 (interim) · Download .docx

Objectives

Write equations of circles and parabolas in standard form.
Identify the center, vertices, foci, and asymptotes of ellipses and hyperbolas.
Classify a conic given its general-form equation.
Complete the square to convert a general equation to standard form.
Graph a conic section from its standard form equation.

Key terms

conic section: Curve formed by a plane cutting a double cone: circle, ellipse, parabola, or hyperbola.
circle: (x−h)² + (y−k)² = r². Center (h, k), radius r.
parabola (vertical): y = a(x−h)² + k or (x−h)² = 4p(y−k). Vertex (h, k); opens up (a>0) or down (a<0).
ellipse: (x−h)²/a² + (y−k)²/b² = 1. Center (h, k); a > b; foci on major axis at distance c = √(a²−b²) from center.
hyperbola (horizontal): (x−h)²/a² − (y−k)²/b² = 1. Opens left-right; asymptotes y − k = ±(b/a)(x − h).
completing the square: Adding (b/2)² to x² + bx to form a perfect square: x² + bx + (b/2)² = (x + b/2)².
eccentricity: e = c/a for ellipse (0 < e < 1) and hyperbola (e > 1); e = 0 for circle; e = 1 for parabola.
discriminant (conics): For Ax² + Bxy + Cy², discriminant B² − 4AC: < 0 and A = C → circle, < 0 → ellipse, = 0 → parabola, > 0 → hyperbola.

The concept

Conic sections are the four curves you get when a flat plane slices a right circular double cone at different angles. Slice parallel to the base: circle. Tilt slightly: ellipse. Tilt more steeply: parabola. Slice through both cones: hyperbola. These are not just geometric curiosities — they describe the paths of planets, the shape of bridges, the reflection of light and sound, and the orbits of satellites.

A circle is all points equidistant from a center. Standard form: (x − h)² + (y − k)² = r². Center is (h, k); radius is r. To write from a general equation, complete the square for x and y separately.

A parabola is all points equidistant from a focus and a directrix. In the precalculus course, the most practical form is the vertex form y = a(x − h)² + k. Vertex is (h, k). If a > 0, the parabola opens up; if a < 0, it opens down. The absolute value of a determines how wide (|a| < 1 = wider, |a| > 1 = narrower).

An ellipse is all points where the sum of distances to two foci is constant. Standard form: (x−h)²/a² + (y−k)²/b² = 1. If a > b, the major axis is horizontal — the ellipse is wider than tall. If b > a, the major axis is vertical. The foci are inside the ellipse at distance c = √(a² − b²) from the center along the major axis.

A hyperbola is all points where the absolute difference of distances to two foci is constant. Two forms: (x−h)²/a² − (y−k)²/b² = 1 (horizontal, opens left-right) and (y−k)²/a² − (x−h)²/b² = 1 (vertical, opens up-down). Key feature: asymptotes. The graph approaches but never reaches y − k = ±(b/a)(x − h) for the horizontal form.

To identify a conic from a general equation Ax² + By² + Cx + Dy + E = 0: if the squared terms have the same coefficient with the same sign → circle (equal) or ellipse (unequal); if the signs differ → hyperbola; if only one squared term → parabola.

Blueprint Pillar 3 — Technology and Society: Ellipses govern Kepler's first law — every planet orbits the Sun in an ellipse with the Sun at one focus. Parabolic dishes focus signals and sound at their focus point — satellite dishes, reflecting telescopes, hearing loops. Hyperbolas appear in GPS triangulation (the set of points with a constant time difference from two receivers forms a hyperbola). Engineering of these shapes requires the precise mathematics of conic sections.

Worked examples

Example 1: Converting to standard form: x² + y² − 6x + 4y − 3 = 0. Group: (x²−6x) + (y²+4y) = 3. Complete the square: (x−3)² − 9 + (y+2)² − 4 = 3. (x−3)² + (y+2)² = 16. Circle, center (3, −2), radius 4.

Example 2: Identifying a hyperbola: 9x² − 4y² = 36. Divide by 36: x²/4 − y²/9 = 1. This is (x−0)²/2² − (y−0)²/3² = 1 — a horizontal hyperbola centered at origin, a = 2, b = 3. Asymptotes: y = ±(3/2)x.

Common mistakes

Adding the wrong completing-the-square constant to both sides. For x² − 6x: add (6/2)² = 9 to BOTH sides of the equation, not just the left side. Forgetting the right side throws off the radius or center.
Confusing a and b in ellipses. a is always the semi-major axis (larger denominator) — not always under x². If the denominator under y² is larger, the major axis is vertical.
Misreading a hyperbola as an ellipse. The key difference: ellipse has + between the squared terms; hyperbola has −. The sign determines the conic, not the shape of the coefficients.

Self-check

Try each one before you look. A miss here costs nothing and tells you exactly what to reread.

1. The equation (x−1)² + (y+3)² = 25 is a circle with radius:

2. For the ellipse (x²/36) + (y²/16) = 1, the semi-major axis length is:

3. Which conic section has a minus sign between its two squared terms?

4. The graph of y = 2(x − 3)² + 1 is a parabola with vertex at:

5. What method converts x² − 4x + y² + 6y = 3 to standard circle form?

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