Conic sections Knowledge-Based Study

Study notes

1. What is a Conic Section?

   A conic section is any curve obtained by intersecting the surface of a double-napped right circular cone with a flat plane. Depending on the tilt of the cutting plane relative to the axis and generators of the cone, four non-degenerate geometric curves can be formed: the circle, the ellipse, the parabola, and the hyperbola.

   The General Second-Degree Equation

   In algebra, every conic section positioned in the Cartesian plane can be expressed by the general second-degree equation in two variables:

   $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

   where $A$, $B$, and $C$ are not all zero.

   Classifying Conics via the Discriminant

   When the conic is oriented such that its axes are parallel to the coordinate axes, the cross-product term is absent ($B = 0$). More generally, the value of the algebraic discriminant, $\Delta = B^2 - 4AC$, classifies the non-degenerate conic as follows:

   • Parabola: $B^2 - 4AC = 0$ (one quadratic term dominates)
   • Ellipse: $B^2 - 4AC < 0$ (both quadratic coefficients have matching signs; if $A = C$ and $B = 0$, it is a circle)
   • Hyperbola: $B^2 - 4AC > 0$ (quadratic coefficients have opposite signs when $B = 0$)

   The Circle

   A circle is the locus of all points in a plane equidistant from a fixed center point $(h, k)$. The constant distance is called the radius, $r$.

   Standard Equation of a Circle

   $$(x - h)^2 + (y - k)^2 = r^2$$

   Expanding this equation yields $x^2 + y^2 + Dx + Ey + F = 0$. Any general equation where $A = C \neq 0$ and $B = 0$ can be transformed into standard form by completing the square on both variables.

   The Parabola

   A parabola is defined as the set of all points $(x, y)$ whose distance to a fixed point (the focus) equals their perpendicular distance to a fixed straight line (the directrix).

   Standard Equations of a Parabola

   • Vertical axis of symmetry: $$(x - h)^2 = 4p(y - k)$$

     The vertex is at $(h, k)$, the focus is at $(h, k + p)$, and the directrix has equation $y = k - p$. If $p > 0$, the parabola opens upward; if $p < 0$, it opens downward.

   • Horizontal axis of symmetry: $$(y - k)^2 = 4p(x - h)$$

     The vertex is at $(h, k)$, the focus is at $(h + p, k)$, and the directrix is $x = h - p$. If $p > 0$, it opens to the right; if $p < 0$, to the left.

   The line segment passing through the focus perpendicular to the axis of symmetry is the latus rectum, with total length $|4p|$.

   The Ellipse

   An ellipse is the set of all points in a plane for which the sum of the distances to two distinct fixed points (the foci) is constant and equal to $2a$, where $2a$ represents the length of the major axis.

   Standard Equation (Center at $(h, k)$)

   • Horizontal major axis: $$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1, \quad a > b > 0$$
   • Vertical major axis: $$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1, \quad a > b > 0$$

   The distance from the center to each focus is $c$, determined by the fundamental relationship:

   $$c^2 = a^2 - b^2$$

   The ratio $e = \frac{c}{a}$ is the eccentricity, satisfying $0 \le e < 1$. A circle corresponds to $e = 0$.

   The Hyperbola

   A hyperbola is the set of all points in a plane such that the absolute difference of the distances to two fixed foci is constant and equal to $2a$.

   Standard Equation (Center at $(h, k)$)

   • Horizontal transverse axis: $$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$$

     Vertices: $(h \pm a, k)$. Foci: $(h \pm c, k)$. Asymptotes: $y - k = \pm \frac{b}{a}(x - h)$.

   • Vertical transverse axis: $$\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$$

     Vertices: $(h, k \pm a)$. Foci: $(h, k \pm c)$. Asymptotes: $y - k = \pm \frac{a}{b}(x - h)$.

   For a hyperbola, the focal distance satisfies $c^2 = a^2 + b^2$, and the eccentricity is strictly greater than 1 ($e = \frac{c}{a} > 1$).

   Example 1: Converting a Parabola to Standard Form

   Consider the equation: $y^2 - 6y - 8x + 25 = 0$.

   1. Isolate terms containing $y$ on the left side: $y^2 - 6y = 8x - 25$.
   2. Complete the square on the left side by adding $\left(\frac{-6}{2}\right)^2 = 9$ to both sides:

      $$y^2 - 6y + 9 = 8x - 25 + 9$$

   3. Factor both sides:

      $$(y - 3)^2 = 8x - 16 \implies (y - 3)^2 = 8(x - 2)$$

   Analysis: This is a horizontal parabola with vertex at $(2, 3)$. Since $4p = 8$, we find $p = 2$. Therefore, the focus is $(2 + 2, 3) = (4, 3)$, and the directrix is $x = 2 - 2 = 0$.

   Example 2: Analyzing an Ellipse

   Consider the equation: $4x^2 + 9y^2 - 16x + 54y + 61 = 0$.

   1. Group the $x$ and $y$ terms: $4(x^2 - 4x) + 9(y^2 + 6y) = -61$.
   2. Complete the square inside each set of parentheses:

      $$4(x^2 - 4x + 4) + 9(y^2 + 6y + 9) = -61 + 4(4) + 9(9)$$

      $$4(x - 2)^2 + 9(y + 3)^2 = -61 + 16 + 81 = 36$$

   3. Divide both sides by 36 to achieve standard form:

      $$\frac{(x - 2)^2}{9} + \frac{(y + 3)^2}{4} = 1$$

   Analysis: This is an ellipse centered at $(2, -3)$ with a horizontal major axis. Here, $a^2 = 9 \implies a = 3$ and $b^2 = 4 \implies b = 2$. Using $c^2 = a^2 - b^2 = 9 - 4 = 5$, the focal distance is $c = \sqrt{5}$. The foci lie at $(2 \pm \sqrt{5}, -3)$.