Revision: Coordinate Geometry >> Straight Lines Mathematics ISC (Commerce) Class 11 CISCE

The slope m of a line is m = tan⁡θ

where θ is the inclination of the line with the positive x-axis.

An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.

The equation of the line with intercepts a and b is \[\frac{x}{a} + \frac{y}{b} = 1\] .Since the line meets the coordinate axes at A and B, the coordinates are A (a, 0) and B (0, b).
Let the given point be P (3, 4).
Here,

\[AP : BP = 2 : 3\]

\[\therefore 3 = \frac{2 \times 0 + 3 \times a}{2 + 3}, 4 = \frac{2 \times b + 3 \times 0}{2 + 3}\]

\[ \Rightarrow 3a = 15, 2b = 20\]

\[ \Rightarrow a = 5, b = 10\]

Hence, the equation of the line is

\[\frac{x}{5} + \frac{y}{10} = 1\]

\[ \Rightarrow 2x + y = 10\]

The two mutually perpendicular number lines intersecting each other at their zeroes are called rectangular axes or coordinate axes, or axes of reference. 

The position of a point in a plane is expressed by a pair of numbers, one concerning the x-axis and the other concerning the y-axis. called co-ordinates. 

• x → distance from y-axis (abscissa)

• y → distance from x-axis (ordinate)

Locus is the path traced by a moving point, which moves so as to satisfy a certain given condition/conditions. 

\[m=\frac{y_2-y_1}{x_2-x_1}\]

When slope and y-intercept are given

y = mx + c​

• m = slope

• c = y-intercept (value of y when x = 0)

When two points are given

\[\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}\]​​

When the slope and one point are given

y − y₁ = m(x − x₁)

Nature of Slope

• m > 0 → rising line

• m < 0 → falling line

• m = 0 → horizontal line

• m = ∞→ vertical line

Parallel Lines 

Two lines are parallel ⇔ , their slopes are equal, m₁ = m₂

​Perpendicular Lines

Two lines are perpendicular ⇔ m₁ × m₂ = −1

​Collinearity of Three Points

Points A, B, and C are collinear

Method 1: Distance method

AB + BC = AC

Method 2: Slope method

Slope of AB = Slope of BC

Sign Convention

• Right of y-axis → +x

• Left of y-axis → −x

• Above x-axis → +y

• Below x-axis → −y

Standard Line Results

• x = 0 → y-axis

• y = 0 → x-axis

• x = a → line parallel to the y-axis

• y = b → line parallel to the x-axis

Quadrant Reminder