8.4 Conic Sections: Parabola, Ellipse & Hyperbola Characteristics

Conic Sections: Parabola, Ellipse & Hyperbola Characteristics

Conic sections constitute the cornerstone of advanced analytic geometry in the Intermediate Mathematics Part II syllabus. On the Pakistan ECAT, conic questions are designed to test algebraic agility, parameter extraction, and geometric classification under strict 60-second time limits. By analyzing conics through the unifying lens of the focus-directrix ratio (eccentricity) and mastering standard canonical orientations, students can bypass multi-line algebraic expansions.

---

1. Unified Geometric Definition and the Role of Eccentricity

A conic section is defined as the locus of a point $P(x, y)$ that moves in a plane such that the ratio of its distance from a fixed point $S$ (the focus) to its perpendicular distance from a fixed straight line $L$ (the directrix) is a constant positive real number $e$ termed the eccentricity:

$\frac{|SP|}{|PM|} = e \iff |SP| = e \cdot |PM|$

Classification by Eccentricity ($e$)

---

2. Algebraic Classification of Conics from the General Second-Degree Equation

Consider the general quadratic equation in two variables:

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

Let $\Delta$ be the determinant matrix representing the system:

$\Delta = \begin{vmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{vmatrix}$

Non-Degenerate Conics ($\Delta \neq 0$)

When $\Delta \neq 0$, the quadratic expression represents an authentic, non-degenerate planar curve whose geometric identity is determined exclusively by the discriminant $B^2 - 4AC$:

1. Parabola: $B^2 - 4AC = 0$
2. Ellipse: $B^2 - 4AC < 0$ (represented as a circle if $A = C$ and $B = 0$)
3. Hyperbola: $B^2 - 4AC > 0$ (represented as a rectangular hyperbola if $A + C = 0$)

Fast Classification for Unrotated Conics ($B = 0$)

When cross-term $B = 0$, the equation simplifies to $Ax^2 + Cy^2 + Dx + Ey + F = 0$. Immediate ECAT classification requires only inspecting the product $A \cdot C$:

• If $A \cdot C = 0$ (one variable is linear while the other is squared): Parabola.
• If $A \cdot C > 0$ with $A = C$: Circle.
• If $A \cdot C > 0$ with $A \neq C$: Ellipse.
• If $A \cdot C < 0$ (opposite algebraic signs): Hyperbola.

---

3. The Parabola

A parabola is the locus of a point whose distance from the focus equals its distance from the directrix ($e = 1$).

Four Standard Orientations with Vertex at Origin $(0, 0)$

Tangents and Normals to $y^2 = 4ax$

1. Point Form: The tangent at point $(x_1, y_1)$ on the parabola is:

$y y_1 = 2a(x + x_1)$

1. Slope Form: The line $y = mx + c$ is tangent to $y^2 = 4ax$ if and only if:

$c = \frac{a}{m} \quad (m \neq 0)$

Yielding the tangent equation:

$y = mx + \frac{a}{m}$

The unique point of contact is $\left( \frac{a}{m^2}, \; \frac{2a}{m} \right)$.

1. Normal Line in Slope Form: The equation of the normal with slope $m$ to $y^2 = 4ax$ is:

$y = mx - 2am - am^3$

---

4. The Ellipse

An ellipse is the locus of a point whose distance from the focus is less than its distance from the directrix ($0 < e < 1$).

Horizontal Standard Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b$)

• Major Axis: Lies along the $x$-axis; total length $= 2a$.
• Minor Axis: Lies along the $y$-axis; total length $= 2b$.
• Fundamental Relationship: $b^2 = a^2(1 - e^2) \implies c^2 = a^2 - b^2$, where $c = ae$.
• Eccentricity Formula:

$e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}} = \frac{\sqrt{a^2 - b^2}}{a}$

• Foci: $(\pm ae, 0) = (\pm c, 0)$.
• Directrices: $x = \pm \frac{a}{e}$.
• Vertices: $(\pm a, 0)$; Co-vertices: $(0, \pm b)$.
• Length of Latus Rectum:

$\text{LR} = \frac{2b^2}{a}$

Focal Distance Property

For any point $P(x, y)$ on the ellipse, the sum of its distances from the two foci $S(ae, 0)$ and $S'(-ae, 0)$ is strictly constant and equal to the major axis length:

$|SP| + |S'P| = 2a$

Vertical Standard Form ($b > a$)

When the major axis lies along the $y$-axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b > a$ (or rewritten as $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$):

• Foci: $(0, \pm be)$ where $e = \sqrt{1 - \frac{a^2}{b^2}}$.
• Directrices: $y = \pm \frac{b}{e}$.
• Length of Latus Rectum: $\frac{2a^2}{b}$.

Tangency Condition for the Ellipse

The line $y = mx + c$ is tangent to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ if and only if:

$c^2 = a^2 m^2 + b^2 \iff c = \pm \sqrt{a^2 m^2 + b^2}$

The tangent equation in slope form is $y = mx \pm \sqrt{a^2 m^2 + b^2}$. The director circle of the ellipse is $x^2 + y^2 = a^2 + b^2$.

---

5. The Hyperbola

A hyperbola is the locus of a point whose distance from the focus exceeds its distance from the directrix ($e > 1$).

Horizontal Standard Form: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• Transverse Axis: Along the $x$-axis; total length $= 2a$.
• Conjugate Axis: Along the $y$-axis; total length $= 2b$.
• Fundamental Relationship: $b^2 = a^2(e^2 - 1) \implies c^2 = a^2 + b^2$, where $c = ae$.
• Eccentricity Formula:

$e = \frac{c}{a} = \sqrt{1 + \frac{b^2}{a^2}} = \frac{\sqrt{a^2 + b^2}}{a} > 1$

• Foci: $(\pm ae, 0) = (\pm c, 0)$.
• Directrices: $x = \pm \frac{a}{e}$.
• Vertices: $(\pm a, 0)$.
• Length of Latus Rectum:

$\text{LR} = \frac{2b^2}{a}$

Focal Distance Property

For any point $P(x, y)$ on the hyperbola, the absolute difference of its distances from the two foci is strictly constant and equal to the transverse axis length:

$\bigl| |SP| - |S'P| \bigr| = 2a$

Asymptotes of the Hyperbola

The asymptotes are straight lines passing through the center to which the branches of the hyperbola approach arbitrarily closely at infinity. Setting the right-hand constant to zero in $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ yields:

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \implies y = \pm \frac{b}{a}x$

Tangency Condition for the Hyperbola

The straight line $y = mx + c$ is tangent to $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if and only if:

$c^2 = a^2 m^2 - b^2 \iff c = \pm \sqrt{a^2 m^2 - b^2} \quad (a^2 m^2 > b^2)$

---

6. Special Hyperbolas: Rectangular & Conjugate

Rectangular (Equilateral) Hyperbola

When the transverse and conjugate semi-axes are equal ($a = b$):

$x^2 - y^2 = a^2$

1. Eccentricity: Universally fixed at $\sqrt{2}$:

$e = \sqrt{1 + \frac{a^2}{a^2}} = \sqrt{1 + 1} = \sqrt{2}$

1. Asymptotes: $y = \pm x$, which are mutually perpendicular ($m_1 m_2 = (1)(-1) = -1$).
2. Rotated Coordinates: Rotating axes by $45^\circ$ yields the canonical reciprocal form:

$xy = c^2$

where the coordinate axes serve as the perpendicular asymptotes.

Conjugate Hyperbola

For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, its conjugate hyperbola is defined by:

$-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \iff \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$

If $e$ is the eccentricity of the primary hyperbola and $e'$ is the eccentricity of its conjugate:

$e = \sqrt{1 + \frac{b^2}{a^2}}, \qquad e' = \sqrt{1 + \frac{a^2}{b^2}}$

They satisfy the celebrated harmonic reciprocal relation:

$\frac{1}{e^2} + \frac{1}{(e')^2} = \frac{a^2}{a^2 + b^2} + \frac{b^2}{a^2 + b^2} = 1$

---

7. Master Comparison of Conic Sections

---

8. Shifted Conics (Translation of Axes)

Replacing $x$ by $x - h$ and $y$ by $y - k$ moves a conic so that its centre (or vertex) is at $(h, k)$ without changing its shape, eccentricity, latus rectum or axis lengths.

Worked example: Find the centre and foci of $9x^2 + 25y^2 - 36x + 150y + 36 = 0$. Complete the squares:

$9(x^2 - 4x) + 25(y^2 + 6y) = -36 \implies 9(x - 2)^2 + 25(y + 3)^2 = -36 + 36 + 225 = 225$

$\frac{(x - 2)^2}{25} + \frac{(y + 3)^2}{9} = 1$

So $a = 5$, $b = 3$, $e = \sqrt{1 - \frac{9}{25}} = \frac{4}{5}$, centre $(2, -3)$, and foci $(2 \pm 4, -3)$, that is $(6, -3)$ and $(-2, -3)$.