# Mathematics（Ⅰ）
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# Unit 4: Straight Lines and Circles

## 4-1 Equation of a Straight Line

### Topic 1: Symmetry in Coordinate Graphs

**(1) Meaning of Slope:**

- Suppose line $L$ is not vertical, and points $A(x_1, y_1)$ and $B(x_2, y_2)$ are two distinct points on $L$.
Then the slope of line $L$ is

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

- When $L$ is a vertical line, we have $x_1 = x_2$, making the denominator 0, so the slope of a vertical line is undefined.

- From a graphical perspective, the slope measures the degree and direction of inclination of the line.

- From a functional perspective, the slope represents the average rate of change of $y$ with respect to $x$, which is constant for a straight line.

**(2) Properties of Slope:**

- The slope of a horizontal line is 0.

- When a line slants upward from the lower left to the upper right, the slope is positive.

- When a line slants downward from the upper left to the lower right, the slope is negative.

- The closer a line is to being vertical, the greater the absolute value of its slope.

- A vertical line has no slope.

|Definition of Slope|Change in Slope|
|--|--|
|  ||

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### Topic 2: Equations of a Straight Line

**(1) Point-Slope Form:**
- A line passing through the point $A(x_0, y_0)$ with a slope of $m$ has the equation

$$y - y_0 = m(x - x_0)$$

**(2) Intercepts:**

- When line $L$ intersects the x-axis at the point $(a, 0)$, $a$ is called the x-intercept of $L$.

- When line $L$ intersects the y-axis at the point $(0, b)$, $b$ is called the y-intercept of $L$.

>Note:
A horizontal line has no x-intercept, and a vertical line has no y-intercept.

**(3) Intercept Form:**
- If $a$ is the x-intercept and $b$ is the y-intercept, and $ab \neq 0$, then the equation of the line is

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

**(4) General Form:**
- For the line $ax + by + c = 0$ with $ab \neq 0$, the slope of line $L$ is $-\frac{a}{b}$

- When $a \neq 0$ and $b = 0$, line $L$ is a vertical line.

- When $a = 0$ and $b \neq 0$, line $L$ is a horizontal line.

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### Topic 3: Translation of a Straight Line

Let $h, k > 0$.

(1) When the line $y = mx$ is translated upward by $k$ units, the new line is $y=mx+k$

(2) When the line $y = mx$ is translated downward by $k$ units, the new line is $y=mx-k$

(3) When the line $y = mx$ is translated to the right by $h$ units, the new line is $y=m(x-h)$

(4) When the line $y = mx$ is translated to the left by $h$ units, the new line is $y=m(x+h)$

>Note: Translation does not change the slope of the line.

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### Topic 4: Relationships Between Two Lines

Let $L_1$ and $L_2$ be two distinct non-vertical lines with slopes $m_1$ and $m_2$, respectively.

(1) If $L_1 // L_2$, then $m_1 = m_2$; and conversely.
(2) If $L_1 \perp L_2$, then $m_1 m_2 = -1$; and conversely.

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