All conic sections share these key elements:

• Focus: A fixed point used to construct and define a conic section
• Directrix: A fixed line used in defining a conic section
• Eccentricity (e): The constant ratio of distance from a moving point to the focus and distance to the directrix
• Vertex: A turning point of any conic section
• Axis: A line which divides a conic into two equal parts (axis of symmetry)
• Centre: The point where the major and minor axes meet (ellipse and hyperbola only)

The value of eccentricity determines the type of conic:

A parabola is the locus of a point that moves such that its distance from a focus equals its distance from a directrix.

Standard Equation (Vertex at Origin)

For a parabola with focus at $(a, 0)$ and directrix $x = -a$:

$y^2 = 4ax$

The focus is $(a, 0)$, directrix is $x = -a$, and vertex is $(0, 0)$.

Other orientations:

• $x^2 = 4by$ (opens upward, focus at $(0, b)$)
• $y^2 = -4ax$ (opens left)
• $x^2 = -4by$ (opens downward)

Translated Parabola

When the vertex is at $(h, k)$:

$(y - k)^2 = 4a(x - h)$

or for parabolas opening upward:

$(x - h)^2 = 4b(y - k)$

Tangent and Normal

For the parabola $y^2 = 4ax$ at point $P(x_1, y_1)$:

• Tangent equation: $yy_1 = 2a(x + x_1)$
• Normal equation: $y_1x + 2ay - 2ay_1 - x_1y_1 = 0$

Parametric Equations

For $y^2 = 4ax$:

$x = at^2, \quad y = 2at$

An ellipse is the locus of a point where the sum of distances from two fixed foci is constant.

Standard Equation (Centre at Origin)

For an ellipse with foci on the x-axis:

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

where $a$ is the semi-major axis, $b$ is the semi-minor axis, and $b^2 = a^2(1 - e^2)$.

• Vertices: $(\pm a, 0)$ or $(0, \pm b)$
• Foci: $(\pm ae, 0)$ or $(0, \pm be)$
• Directrices: $x = \pm \frac{a}{e}$ or $y = \pm \frac{b}{e}$

Translated Ellipse

When the centre is at $(h, k)$:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

Tangent and Normal

For ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at $P(x_1, y_1)$:

• Tangent equation: $\frac{x_1x}{a^2} + \frac{y_1y}{b^2} = 1$
• Normal equation: $\frac{y_1(x-x_1)}{b^2} = \frac{x_1(y-y_1)}{a^2}$

Parametric Equations

$x = a\cos\theta, \quad y = b\sin\theta$

A hyperbola is the locus of a point where the difference of distances from two foci is constant.

Standard Equation (Centre at Origin)

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

where $b^2 = a^2(e^2 - 1)$.

• Vertices: $(\pm a, 0)$
• Foci: $(\pm ae, 0)$
• Directrices: $x = \pm \frac{a}{e}$
• Asymptotes: $y = \pm \frac{b}{a}x$

Translated Hyperbola

When centre is at $(h, k)$:

$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

The asymptotes become: $y - k = \pm \frac{b}{a}(x - h)$

Tangent and Normal

For hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ at $P(x_1, y_1)$:

• Tangent equation: $\frac{x_1x}{a^2} - \frac{y_1y}{b^2} = 1$
• Normal equation: $\frac{a^2}{x_1}x + \frac{b^2}{y_1}y = a^2 + b^2$

Parametric Equations

$x = a\sec\theta, \quad y = b\tan\theta$

Polar coordinates represent points using a distance $r$ from a fixed point (pole) and an angle $\theta$ from a fixed direction (polar axis).

Relationship with Rectangular Coordinates

From the right triangle in Figure:

$x = r\cos\theta, \quad y = r\sin\theta$

$x^2 + y^2 = r^2, \quad \tan\theta = \frac{y}{x}$

Converting Polar to Rectangular

Example: Find rectangular coordinates of $P(5, \frac{\pi}{2})$

Solution: $x = 5\cos(\frac{\pi}{2}) = 5(0) = 0$ $y = 5\sin(\frac{\pi}{2}) = 5(1) = 5$

Rectangular coordinates: $P(0, 5)$

Converting Rectangular to Polar

Example: Find polar coordinates of $(2, 2)$

Solution: $r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$ $\tan\theta = \frac{2}{2} = 1 \Rightarrow \theta = \frac{\pi}{4}$

Polar coordinates: $(2\sqrt{2}, \frac{\pi}{4})$

Graphs of Polar Equations

Key polar curve types:

• Circles: $r = a\sin\theta$, $r = a\cos\theta$, $r = a$
• Limaçons: $r = a \pm b\cos\theta$, $r = a \pm b\sin\theta$
• Lemniscates: $r^2 = a^2\cos 2\theta$
• Roses: $r = a\sin n\theta$, $r = a\cos n\theta$ (n petals if n is odd, 2n if even)

In Tanzania, parabolic reflectors are commonly used in satellite TV dishes (like those installed on many households for DSTV or Zuku). The dish is shaped as a parabola with the receiver positioned at the focus, allowing signals to be concentrated for clear reception. Similarly, elliptical arches are used in bridge construction across the country (such as the Mkumbara Bridge in Lindi), where the ellipse's property of having a constant sum of distances from two foci provides structural stability. Understanding polar coordinates helps in navigation systems used by daladala drivers and in surveying land for farming plots in rural areas like those in Morogoro.