Chapter 9: Straight Lines | SJMaths Class 11

1. Slope (Gradient)

The slope (\(m\)) of a line describes its steepness and direction.

If inclination is \( \theta \), then \( m = \tan \theta \).
If passing through \( (x_1, y_1) \) and \( (x_2, y_2) \), then \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Slope Detective

Enter two points to find the slope and angle.

2. Line Builder Workshop

Generate equations using different available data.

Slope-Intercept Form

\( y = mx + c \)

Point-Slope Form

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

Two-Point Form

📌 Intercept Form of a Line

If a line cuts x-axis at \( a \) and y-axis at \( b \), then its equation is:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

Exam Insight: Used when intercepts are given directly in the question.

3. General Equation & Distance

General form: \( Ax + By + C = 0 \).

Parameter	Formula
Slope (m)	\( -A/B \)
X-intercept	\( -C/A \)
Y-intercept	\( -C/B \)

👨‍👩‍👧‍👦 Family of Lines

1. Lines through intersection of:

\( L_1 = 0 \) and \( L_2 = 0 \)

General family: \( L_1 + \lambda L_2 = 0 \)

2. Lines parallel to:

\( Ax + By + C = 0 \)

Family: \( Ax + By + k = 0 \)

CBSE Insight: These questions test concept, not calculation.

Distance Calculator

Distance from point \( P(x_1, y_1) \) to line \( Ax + By + C = 0 \).

4. Parallel vs Perpendicular

Relation Checker

Enter slopes \( m_1 \) and \( m_2 \).

📐 Angle Between Two Lines

If slopes of two lines are \( m_1 \) and \( m_2 \), then angle \( \theta \) between them is:

\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

If \( m_1 m_2 = -1 \) → lines are perpendicular
If \( m_1 = m_2 \) → lines are parallel

CBSE Tip:
Questions often ask to find angle OR prove lines are perpendicular using this formula.

⚠️ Common Mistakes to Avoid

✘ Using wrong slope formula
✘ Forgetting negative sign in slope \( -A/B \)
✘ Confusing distance formula with point distance
✘ Not checking perpendicular condition properly
✘ Writing equation in wrong form

Concept Mastery Quiz

1. Slope of line parallel to X-axis is:

2. If two lines are perpendicular, product of slopes is:

3. Distance of (0,0) from 3x + 4y = 10 is:

4. Equation of X-axis is:

5. Angle between lines with slopes 1 and -1 is:

🧠 One-Page Revision Checklist

✔ Slope formulas revised
✔ Angle of inclination understood
✔ All forms of line equations revised
✔ Distance formula memorised
✔ Parallel & perpendicular conditions clear
✔ Angle between lines formula revised
✔ Family of lines concept clear

Self-Check:

❓ Find equation of line parallel to y-axis
❓ Find angle between slopes 2 and −1/2
❓ Distance of (1,2) from 3x−4y+5=0