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Geometry and Algebra

Analytic geometry in the plane: lines and conics

Lines in parametric and Cartesian form, parallelism and point-line distance, ellipse, hyperbola and parabola as loci, classifying conics with eigenvalues.

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Analytic geometry turns geometric problems into equations: a point is a pair of numbers, a line a first-degree equation, a conic a second-degree equation. In this lesson we cover lines in the plane (equations, parallelism, distances) and conics, with their definitions as loci and the classification that uses eigenvalues.

Lines in the plane

A line can be described in two ways.

  • Parametric form: through P_0 = (x_0, y_0) with direction vector \mathbf d = (l, m): (x, y) = (x_0, y_0) + t\,(l, m).
  • Cartesian form: ax + by + c = 0. The vector \mathbf n = (a, b) is perpendicular to the line.

Why (a, b) is normal

If P_1 and P_2 are on the line, ax_1 + by_1 + c = 0 and ax_2 + by_2 + c = 0. Subtracting, a(x_2 - x_1) + b(y_2 - y_1) = 0, that is \mathbf n\cdot\overrightarrow{P_1P_2} = 0: the vector (a, b) is orthogonal to every vector along the line. A direction vector is therefore (-b, a), and the slope is m = -a/b.

Relative position. Two lines are parallel if their normal vectors are proportional, perpendicular if the normals are orthogonal (a_1a_2 + b_1b_2 = 0, or with slopes, m_1 m_2 = -1). Otherwise they meet at one point, found by solving the system: the lesson on linear systems seen through geometry.

Distance from a point to a line

    \[ d(Q, r) = \frac{|a x_Q + b y_Q + c|}{\sqrt{a^2 + b^2}} \]

Proof. Let P_0 be any point on the line. The distance is the magnitude of the projection of \overrightarrow{P_0Q} onto the normal direction \mathbf n:

    \[ d = \frac{|\overrightarrow{P_0Q}\cdot\mathbf n|}{|\mathbf n|} = \frac{|a(x_Q - x_0) + b(y_Q - y_0)|}{\sqrt{a^2 + b^2}} = \frac{|ax_Q + by_Q + c|}{\sqrt{a^2 + b^2}} \]

because -ax_0 - by_0 = c. Example: the distance of (3, 1) from 4x - 3y + 1 = 0 is |12 - 3 + 1|/5 = 2.

Conics as loci

Ellipse, hyperbola and parabola with foci, directrix and their defining property
Ellipse: the sum of the distances from two foci is constant. Hyperbola: the absolute value of the difference is constant. Parabola: every point is equidistant from the focus F and the directrix d.

Circle. The points at distance r from the centre (x_0, y_0): (x - x_0)^2 + (y - y_0)^2 = r^2.

Ellipse. The points with PF_1 + PF_2 = 2a. With foci at (\pm c, 0) and b^2 = a^2 - c^2:

    \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

The eccentricity e = c/a lies between 0 (circle) and 1. The Earth’s orbit has e \approx 0.017, almost a circle; Halley’s comet e \approx 0.97.

Hyperbola. The points with |PF_1 - PF_2| = 2a. With foci at (\pm c, 0) and b^2 = c^2 - a^2:

    \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\qquad \text{asymptotes } y = \pm\frac ba x \]

The LORAN navigation system used hyperbolas: the difference in arrival times of the signals from two stations places the receiver on a hyperbola with the stations as foci.

Parabola. The points equidistant from the focus F = (0, p) and the directrix y = -p:

    \[ x^2 + (y - p)^2 = (y + p)^2 \quad\Rightarrow\quad y = \frac{x^2}{4p} \]

Its optical property is famous: rays parallel to the axis are all reflected into the focus. That is why satellite dishes have the receiver (LNB) at the focus, and car headlights have the bulb at the focus of a parabolic mirror.

Proof: the equation of the ellipse

With F_{1,2} = (\mp c, 0), the condition is \sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a. Isolating one root and squaring:

    \[ (x + c)^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + (x - c)^2 + y^2 \quad\Rightarrow\quad a\sqrt{(x - c)^2 + y^2} = a^2 - cx \]

Squaring again and simplifying: (a^2 - c^2)x^2 + a^2y^2 = a^2(a^2 - c^2). With b^2 = a^2 - c^2 and dividing by a^2b^2 you get the standard equation.

Classifying conics with eigenvalues

A general second-degree equation

    \[ ax^2 + 2bxy + cy^2 + dx + ey + f = 0 \]

can be written with a symmetric matrix: the quadratic part is \mathbf x^TA\mathbf x with A = \begin{pmatrix}a & b\\ b & c\end{pmatrix}. By the spectral theorem there is a rotation of the axes that makes A diagonal: in the new frame the mixed term xy disappears and the quadratic part becomes \lambda_1x'^2 + \lambda_2y'^2. The type of conic depends on the signs of the eigenvalues, that is on the determinant:

  • \det A = ac - b^2 \gt 0 (eigenvalues of the same sign): ellipse;
  • \det A \lt 0 (opposite signs): hyperbola;
  • \det A = 0 (one zero eigenvalue): parabola.

(Apart from degenerate cases, such as pairs of lines or a single point, which are recognised from the determinant of the full 3×3 matrix.)

Example. 5x^2 + 4xy + 2y^2 = 6. A = \begin{pmatrix}5 & 2\\ 2 & 2\end{pmatrix}, characteristic polynomial \lambda^2 - 7\lambda + 6, eigenvalues 6 and 1. In the eigenvector frame, (2, 1)/\sqrt5 for 6 and (-1, 2)/\sqrt5 for 1, the equation is 6x'^2 + y'^2 = 6, that is x'^2 + y'^2/6 = 1: an ellipse with semi-axes 1 and \sqrt6, rotated by \arctan(1/2) \approx 26.6°.

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