ICSE Class 10 Maths Chapter 14 Equation of a Straight Line

Chapter 14

Equation of a Straight Line

Points To Remember

1 Inclination of a Line

The angle of inclination or simply an inclination of a line is the angle θ which the part of the line above x-axis makes with the positive direction of x-axis and measured in anticlockwise direction.

Two diagrams show coordinate axes with angle θ marked between a line and the positive x-axis direction.

Remarks:

(i) The inclination of x-axis is 0°.

(ii) The inclination of every line parallel to x-axis is 0°.

(iii) The inclination of y-axis is 90°.

(iv) The inclination of every line parallel to y-axis is 90°.

Horizontal Line: Any line parallel to x-axis, is called a horizontal line.

Vertical Line: Any line parallel to y-axis, is called a vertical line.

Oblique Line: A line which is neither parallel to x-axis nor parallel to y-axis, is called an oblique line.

2. Slope or Gradient of a Line

If θ is the inclination of a line, then the value of tan θ is called the slope of that line and it is denoted by m. i.e. m = tan θ.

3. Some Results on Slope of a Line

Theorem 1. (i) The slope of a line parallel to x-axis is 0.

(ii) The slope of x-axis is 0.

Proof. (i) We know that the inclination of a line parallel to x-axis is 0°.

Therefore, Slope of a line parallel to x-axis = tan 0° = 0.

(ii) We know that the inclination of x-axis is 0°.

Therefore, Slope of x-axis = tan 0° = 0.

Remark: The slope of a horizontal line is 0.

Theorem 2. (i) The slope of y-axis is not defined.

(ii) The slope of any line parallel to y-axis, is not defined.

Proof. (i) We know that the inclination of y-axis is 90°.

Therefore, Slope of y-axis = tan 90° = ∞, which is not defined.

(ii) We know that the inclination of a line parallel to y-axis is 90°.

Therefore, Slope of a line parallel to y-axis = tan 90° = ∞, which is not defined.

Remark: The slope of a vertical line is not defined.

Theorem 3. Two non-vertical lines are parallel if and only if their slopes are equal.

Proof. Let l₁ and l₂ be two non-vertical lines with slopes m₁ and m₂ respectively and inclinations θ₁ and θ₂ respectively.

Then m₁ = tan θ₁ and m₂ = tan θ₂.

Let l₁ ∥ l₂. Then,

l₁ ∥ l₂ \(\Rightarrow\) θ₁ = θ₂ [Corresponding angles]

\(\Rightarrow\) tan θ₁ = tan θ₂

\(\Rightarrow\) m₁ = m₂.

Conversely, let m₁ = m₂. Then,

m₁ = m₂ \(\Rightarrow\) tan θ₁ = tan θ₂

\(\Rightarrow\) θ₁ = θ₂

\(\Rightarrow\) l₁ ∥ l₂ [∴ θ₁ and θ₂ are corresponding angles]

Therefore, l₁ ∥ l₂ \(\Leftrightarrow\) m₁ = m₂.

Hence, two lines are parallel if and only if their slopes are equal.

A diagram shows two parallel lines with angles θ₁ and θ₂ marked with the x-axis.

Theorem 4. Two non-vertical lines with slopes m₁ and m₂ are perpendicular to each other, if and only if m₁ m₂ = - 1.

Proof. Let l₁ and l₂ be two non-vertical lines with slopes m₁ and m₂ respectively and inclinations θ₁ and θ₂ respectively. Then,

m₁ = tan θ₁ and m₂ = tan θ₂.

Let l₁ ⊥ l₂. Then,

l₁ ⊥ l₂ \(\Rightarrow\) θ₂ = (90° + θ₁)

\(\Rightarrow\) tan θ₂ = tan (90° + θ₁) = - cot θ₁

\(\Rightarrow\) tan θ₂ = \(-\frac{1}{\tan θ₁}\)

\(\Rightarrow\) tan θ₁ \(\cdot\) tan θ₂ = - 1

\(\Rightarrow\) m₁m₂ = - 1.

Conversely, let m₁m₂ = - 1. Then,

m₁m₂ = - 1 \(\Rightarrow\) tan θ₁ \(\cdot\) tan θ₂ = - 1

\(\Rightarrow\) tan θ₂ = \(-\frac{1}{\tan θ₁}\)

A diagram shows two perpendicular lines with angles θ₁ and θ₂ marked, with a 90° angle indicated between them.

Teacher's Note

Understanding slopes helps you determine if roads or ramps are parallel or perpendicular, which is essential in city planning and architecture.

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