Quick Revision Notes: Straight Lines

Class 11 Mathematics (MP Board)

1. Introduction & Basic Concepts

• Straight Line: A locus of a point which moves in a constant direction. Its general equation is a linear equation in two variables: $Ax + By + C = 0$.
• Slope (Gradient) of a Line ($m$): If a line makes an angle $\theta$ with the positive direction of the X-axis measured in anti-clockwise direction, then its slope is given by: $$m = \tan\theta \quad (\text{where } \theta \neq 90^\circ)$$
• Slope from Two Points: If a line passes through points $P(x_1, y_1)$ and $Q(x_2, y_2)$, then: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

2. Condition for Parallel and Perpendicular Lines

Let $m_1$ and $m_2$ be the slopes of two non-vertical lines:

• Parallel Lines: Two lines are parallel if and only if their slopes are equal. $$m_1 = m_2$$
• Perpendicular Lines: Two lines are perpendicular if and only if the product of their slopes is $-1$. $$m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$$

3. Angle Between Two Lines

The acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by: $$\tan\theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$$

4. Forms of the Equation of a Line

• Slope-Intercept Form: $$y = mx + c$$ (where $m$ = slope, $c$ = Y-intercept)

• Point-Slope Form: $$y - y_1 = m(x - x_1)$$ (passing through point $(x_1, y_1)$ with slope $m$)

• Two-Point Form: $$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$$

• Intercept Form: $$\frac{x}{a} + \frac{y}{b} = 1$$ (where $a$ = X-intercept, $b$ = Y-intercept)

• Normal Form: $$x \cos\alpha + y \sin\alpha = p$$ (where $p$ = perpendicular distance from origin, $\alpha$ = angle of the perpendicular with the positive X-axis)

5. General Equation of a Line

The general linear equation of a straight line is represented as: $$Ax + By + C = 0$$

• Slope of the line ($m$): $-\frac{\text{Coefficient of } x}{\text{Coefficient of } y} = -\frac{A}{B}$
• X-intercept: $-\frac{C}{A}$
• Y-intercept: $-\frac{C}{B}$

6. Distance Formulas

• Distance of a Point from a Line: The perpendicular distance ($d$) of a point $P(x_1, y_1)$ from the line $Ax + By + C = 0$ is: $$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$

• Distance Between Two Parallel Lines: The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is: $$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$

Important Tips for MP Board Exam:

1. Always write the formula before substituting values in numerical problems.
2. Don't forget to write units (like 'sq. units' or 'units') in distance and area problems.
3. Be careful with signs (+/-) while calculating slopes and intercepts from the general equation.