Straight Lines — Complete Formula

Basic Concepts

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Slope (General)	[m = \tan\theta]	[m] = slope, [\theta] = angle with +x-axis	Inclination of line ⭐
Slope (Two Points)	[m = \dfrac{y_2 – y_1}{x_2 – x_1}]	[(x_1, y_1), (x_2, y_2)] = two points	Very important ⭐

💡 Memory Hint:
Slope = rise/run

Concept	Formula	Symbols Meaning	Key Notes
Equation	[y = mx + c]	[c] = y-intercept	Most used ⭐

Concept	Formula	Symbols Meaning	Key Notes
Equation	[y – y_1 = m(x – x_1)]	[(x_1, y_1)] = point	Useful in JEE ⭐

Concept	Formula	Symbols Meaning	Key Notes
Equation	[\dfrac{y – y_1}{y_2 – y_1} = \dfrac{x – x_1}{x_2 – x_1}]	[(x_1, y_1), (x_2, y_2)]	Direct use ⭐

Concept	Formula	Symbols Meaning	Key Notes
Equation	[\dfrac{x}{a} + \dfrac{y}{b} = 1]	[a] = x-intercept, [b] = y-intercept	Very important ⭐

Concept	Formula	Symbols Meaning	Key Notes
Equation	[x\cos\alpha + y\sin\alpha = p]	[p] = perpendicular distance from origin, [\alpha] = angle of normal	Advanced ⭐

Concept	Formula	Symbols Meaning	Key Notes
General Equation	[Ax + By + C = 0]	[A, B, C] = constants	Most general ⭐
Slope	[m = -\dfrac{A}{B}]	—	Derived relation ⭐

💡 Memory Hint:
Convert any line → general form

Concept	Formula	Symbols Meaning	Key Notes
Angle	[\tan\theta = \dfrac{m_1 – m_2}{1 + m_1 m_2}]

Concept	Formula	Symbols Meaning	Key Notes
Parallel	[m_1 = m_2]	—	Same slope ⭐
Perpendicular	[m_1 m_2 = -1]	—	Product = -1 ⭐

💡 Memory Hint:
Perpendicular → negative reciprocal

Concept	Formula	Symbols Meaning	Key Notes
Distance	[d = \dfrac{Ax_1 + By_1 + C}{\sqrt{A^2 + B^2}}]

💡 Memory Hint:
Always take absolute value ⚠️

Concept	Formula	Symbols Meaning	Key Notes
Foot Coordinates	[\left(x – \dfrac{A(Ax + By + C)}{A^2 + B^2}, ; y – \dfrac{B(Ax + By + C)}{A^2 + B^2}\right)]	[(x, y)] = given point	Advanced ⭐

Concept	Formula	Symbols Meaning	Key Notes
Distance	[d = \dfrac{C_1 – C_2}{\sqrt{A^2 + B^2}}]

Concept	Formula	Symbols Meaning	Key Notes
Through Intersection	[L_1 + \lambda L_2 = 0]	[\lambda] = parameter	Very useful ⭐

💡 Memory Hint:
Used in pair of lines problems

Concept	Formula	Symbols Meaning	Key Notes
Angle Bisector	[\dfrac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \dfrac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}}]	—	Very important ⭐

Concept	Formula	Symbols Meaning	Key Notes
Image Formula	Use reflection via foot of perpendicular	—	Common in JEE ⭐

💡 Memory Hint:
Image = equal distance on opposite side

Concept	Formula	Symbols Meaning	Key Notes
Horizontal Line	[y = c]	—	Slope = 0
Vertical Line	[x = c]	—	Slope undefined ⚠️

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