Coordinate geometry: equations of lines and curves, parametric equations

Coordinate Geometry: Lines and Curves

1. Lines – The Straight Roads of the Plane 🚗

A line is like a straight road that never ends. In the coordinate plane we can describe it with a simple equation. The most common form is the slope‑intercept form:

$y = mx + c$

• $m$ = slope (how steep the road is)
• $c$ = y‑intercept (where the road crosses the y‑axis)

If you know two points on the road, you can find the slope with the “rise over run” formula:

$m = \dfrac{y_2-y_1}{x_2-x_1}$

Once you have the slope, plug it into the point‑slope form to write the line’s equation:

$y - y_1 = m(x - x_1)$

2. Parallel & Perpendicular Lines 🔄

• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$.

Example: Find the equation of a line parallel to $y = 2x + 3$ that passes through $(4, 1)$.

1. Parallel slope: $m = 2$.
2. Use point‑slope: $y - 1 = 2(x - 4)$.
3. Simplify: $y = 2x - 7$.

3. Standard Form of a Line ➗

$Ax + By = C$ where $A$, $B$, $C$ are integers and $A \ge 0$. This form is handy for quick calculations and for checking if two lines intersect at integer points.

4. Curves – The Curvy Roads 🛤️

4.1 Circle

A circle is all points at a fixed distance (radius $r$) from a centre $(h,k)$.

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

Example: Circle with centre $(1, -2)$ and radius $5$:

$(x-1)^2 + (y+2)^2 = 25$

4.2 Parabola

A parabola opens either up/down (vertical) or left/right (horizontal). The standard vertical form is:

$y = ax^2 + bx + c$

If the axis of symmetry is vertical, the vertex form is useful:

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

4.3 Ellipse & Hyperbola

Ellipse (oval shape):

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

Hyperbola (two separate curves):

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

5. Parametric Equations – GPS Coordinates 📍

Instead of one equation, a parametric form gives two equations that describe the same curve using a parameter $t$ (think of $t$ as “time” or a “step” along the path).

$x = f(t), \quad y = g(t)$

5.1 Parametric Line

A straight line can be written as:

$x = x_0 + at, \quad y = y_0 + bt$

Here $(x_0, y_0)$ is a point on the line and $(a,b)$ is a direction vector. The slope is $m = \dfrac{b}{a}$ (if $a eq 0$).

5.2 Parametric Circle

A circle centred at $(h,k)$ with radius $r$:

$x = h + r\cos t, \quad y = k + r\sin t$

As $t$ goes from $0$ to $2\pi$, you trace the whole circle once.

5.3 Parametric Ellipse

Ellipse with semi‑axes $a$ and $b$:

$x = h + a\cos t, \quad y = k + b\sin t$

6. Working with Parametric Curves – Finding the Slope 🧮

To find the slope of a curve at a particular $t$, differentiate both $x$ and $y$ with respect to $t$ and divide:

$\displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

Example: For the circle $x = \cos t$, $y = \sin t$, at $t = \frac{\pi}{4}$:

1. $dx/dt = -\sin t$, $dy/dt = \cos t$.
2. At $t = \frac{\pi}{4}$, $dx/dt = -\frac{\sqrt{2}}{2}$, $dy/dt = \frac{\sqrt{2}}{2}$.
3. $\displaystyle \frac{dy}{dx} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1$.

7. Quick Reference Tables 📚

8. Practice Problems 🎯

1. Find the equation of the line passing through $(2,3)$ and $(5,11)$.
2. Write the equation of a circle with centre $(0,0)$ and radius $4$ in parametric form.
3. Determine the slope of the ellipse $x^2/9 + y^2/4 = 1$ at the point $(3,0)$ using parametric equations.
4. Show that the lines $y = 2x + 1$ and $y = -\frac{1}{2}x + 4$ are perpendicular.

Remember: Think of lines as roads, curves as scenic routes, and parametric equations as GPS coordinates that guide you along the path. Happy exploring! 🚀