【Math C】Curves on a Plane 〜Welcome to the World of Beautiful Curves!〜
Hello everyone! From today, we will start learning the chapter on "Curves on a Plane."
You might be thinking, "I only know circles and parabolas (\(y=ax^2\))..." But once you master this chapter, you will be able to explain all the "beautiful shapes" in the world—from the orbits of planets in space to the mechanisms of satellite dishes—using mathematical formulas!
It might look a bit complicated at first, but don't worry. If you organize the characteristics of each shape one by one like a puzzle, you will definitely be able to understand them. Let’s take it one step at a time, and don't rush!
1. Conic Sections (1): Parabolas
You are already familiar with parabolas from junior high and high school math, but here we will define them from a slightly different perspective. In Math C, we consider a parabola as a collection of points that are equidistant from a "point (focus)" and a "line (directrix)."
Standard Form of a Parabola
The equation of a parabola with focus \(F(p, 0)\) and directrix \(x = -p\) is:
\(y^2 = 4px\)
- Focus: \(F(p, 0)\) … Imagine this as the point where radio waves gather.
- Directrix: \(x = -p\) … The baseline "wall."
- Vertex: The origin \((0, 0)\)
Point: Which way does it open?
If it's in the form \(y^2 = 4px\), it opens horizontally (along the \(x\)-axis). If it's in the form \(x^2 = 4py\), it opens vertically (along the \(y\)-axis—the familiar shape you already know).
Fun Fact: The "parabola" in a parabolic antenna comes from the English word "parabola." It has the property of reflecting waves from far away and collecting them all at the "focus!"
Common Mistake
When you see the equation \(y^2 = 8x\), it's easy to make a mistake with the value of \(p\). Since \(4p = 8\), \(p=2\). Don't forget the "4" in the formula!
2. Conic Sections (2): Ellipses
Simply put, an ellipse is a "flattened circle." It is a collection of points where the sum of the distances from two points is constant.
Standard Form of an Ellipse
\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (\(a > 0, b > 0\))
- When \(a > b\): A horizontally elongated ellipse. The foci are on the \(x\)-axis.
- When \(a < b\): A vertically elongated ellipse. The foci are on the \(y\)-axis.
- How to find the foci (when \(a > b\)): Use \(c\) where \(c^2 = a^2 - b^2\); the foci are \((\pm c, 0)\).
Understand with an Image!
If you stick two pins into a board, loop a string around them, and pull the string taut with a pencil, you can draw an "ellipse." These two pins are the "foci." Since the length of the string doesn't change, the "sum of the distances" remains constant.
Summary: What determines the shape of an ellipse?
Compare \(a^2\) under \(x\) and \(b^2\) under \(y\)!
The longer axis is called the major axis, and the shorter one is called the minor axis.
3. Conic Sections (3): Hyperbolas
As the name suggests, it consists of two opposing curves. This time, it is a collection of points where the difference of the distances from two points is constant.
Standard Form of a Hyperbola
\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) … Opens left and right
\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1\) … Opens up and down
- Asymptotes: Lines that the curve approaches indefinitely but never touches. The equations are \(y = \pm \frac{b}{a}x\).
- How to find the foci: Use \(c\) where \(c^2 = a^2 + b^2\). While the ellipse used a minus sign, the hyperbola uses a plus sign!
Tips for Memorization
Remember: Ellipses are about "sum," so the formula has a "+," while hyperbolas are about "difference," so the formula has a "−."
Also, when drawing a hyperbola, it's easier to draw the asymptotes with dashed lines first, then sketch the curves along them.
4. Parametric Representation
Until now, we have thought about curves using relationships between \(x\) and \(y\) (like \(y=f(x)\)). Here, we use a third variable, \(t\) or \(\theta\), to express \(x\) and \(y\).
Representative Examples
- Circle (\(x^2 + y^2 = r^2\)):
\(x = r \cos \theta\)
\(y = r \sin \theta\) - Ellipse (\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)):
\(x = a \cos \theta\)
\(y = b \sin \theta\)
Analogy: It’s like \(x\) and \(y\) are not talking directly, but communicating through a mutual friend, \(t\) (the parameter). As \(t\) moves, the positions of \(x\) and \(y\) are determined accordingly.
Step: How to eliminate the parameter
1. Rewrite as \(\cos \theta = \frac{x}{a}\) and \(\sin \theta = \frac{y}{b}\).
2. Substitute them into \(\sin^2 \theta + \cos^2 \theta = 1\).
By doing this, you can return to the familiar equation involving only \(x\) and \(y\)!
5. Polar Coordinates and Polar Equations
Finally, let's learn a "new rule" for expressing location. Instead of the usual \(x, y\) coordinates (Cartesian coordinates) like "how far right and how far up," we specify the location by "in what direction and how far."
Basics of Polar Coordinates
Point \(P\) is represented as \((r, \theta)\).
\(r\): The distance from the origin (the pole).
\(\theta\): The angle (argument) from the initial line (the positive \(x\)-axis).
Coordinate Conversion Formulas
This conversion is the very definition of trigonometric functions!
\(x = r \cos \theta\)
\(y = r \sin \theta\)
For the reverse conversion, we use \(r^2 = x^2 + y^2\) or \(\tan \theta = \frac{y}{x}\).
Polar Equations
An equation expressed as a relationship between \(r\) and \(\theta\) is called a polar equation.
For example, \(r = a\) (constant) represents a "circle centered at the pole with radius \(a\)." Simple, right?
Fun Fact: Radar and fish finders use this "polar coordinate" system. They emit radio waves while rotating from the center, identifying the location of targets based on the distance \(r\) and the angle \(\theta\).
Summary: Key points to survive this chapter
You might feel overwhelmed by the number of formulas at first, but the most important thing is to "draw figures and visualize them."
★ Today's Summary:
1. Parabola: \(y^2 = 4px\) (Equidistant from a focus and directrix)
2. Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (Constant sum of distances)
3. Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) (Constant difference of distances)
4. Parametric: Express \(x, y\) using a third variable. Make full use of trigonometric identities!
5. Polar Coordinates: Represent locations using distance \(r\) and angle \(\theta\).
When you encounter a difficult problem, try going back to the basic definitions. "It may feel difficult at first, but it will be fine." As you solve more practice problems, it will start to feel like solving a puzzle. I’m rooting for you!