Equation of Straight Line - Practice Questions & MCQ

Equation of Straight Line

(a) Slope-Intercept form

Consider the given figure

AB is a straight line with slope m and intercept c on Y-axis. P(x, y) any point on the straight line. PL is perpendicular to X-axis and MQ is perpendicular to Y-axis

The equation of a straight line whose slope is given as m and making y-intercept of length c unit is y = mx + c. 

If the straight line passing through the origin, then equation of straight line become y = mx

If Equation of straight line is Ax + By + C = 0, then

We can write By = -Ax - C

(b) Point-Slope form

Let the equation of give line l with slope ‘m’ is 

y = mx + c    …..(i) 

(x₁,y₁) lies on the line i

y₁= mx₁+c   ……(ii)

From (i) and (ii) [(ii) - (i)]

y - y₁ = m( x - x₁)

The equation of a straight line whose slope is given as ‘m’ and passes through the point (x₁,y₁) is  .

Equation of Straight Line

(c) Two-point form

The equation of a straight line passing through the two given points (x₁,y₁) and  (x₂,y₂) is  given by

.

Proof:

Let the equation of straight line l with slope ‘m’ be

y = mx + c                 ……(i)

Points (x₁,y₁) and  (x₂,y₂) pass through the given line l,  then

y₁ = mx₁ + c   ……(ii)

y₂ = mx₂ + c   ……(iii)

Subtract eq (ii) from eq (i)

y - y₁ = m( x - x₁ )   ……(iv)

Subtract eq (iii) from eq (i)

y - y₂ = m( x - x₂ )   ……(v)

Divide eq (iv) by eq (v)

Determinant Form: 

(d) Intercept form of line

Equation of a straight line which makes intercepts ‘a’ and ‘b’ on X-axis and Y-axis respectively is given by

Proof:

A straight line which cut X-axis at A (a, 0) and Y-axis at B (0, b)

Using the concept of two points form of a line

Equation of a straight line through the two-point A(a, 0) and B(0, b)

Note:

For the general equation Ax + By + C = 0

If C ≠ 0, then Ax + By + C = 0 can be written as

Hence, x-intercept is   and y-intercept is .

Normal and Parametric form of a line

Normal form of line

Equation of straight line on which the length of the perpendicular from the origin is p and this normal makes an angle θ with the positive direction of X-axis is given by

Proof:

AB is the straight line and length of perpendicular from origin to the line is p (i.e. ON = p).

Line AB cuts X-axis and Y-axis at point Q and R respectively 

Using Slope-point form, equation of line AB is

Parametric form of a line

The equation of a straight line passing through the point (x₁,y₁) and making an angle θ with the positive direction of X-axis is 

Where r is the directed distance between the points (x, y) and (x₁,y₁).

Proof:

AB is a straight line passing through the point P(x₁,y₁) and meets X-axis at R and makes an angle θ with the positive direction of X-axis.

Let Q(x, y) be any point on the line AB at a distance 'r' from P

As from the figure

From the above two equation