EQUATION OF THE LINE PARALLEL TO X AND Y AXIS

Example 1 :

Find the equation of a straight line passing through the mid-point of a line segment joining the points (1, -5) , (4, 2) and parallel to :

(i) X axis   (ii) Y axis

Solution :

Midpoint of the line segment  =  (x₁ + x₂)/2, (y₁ + y₂)/2

  =  (1 + 4)/2 , (-5 + 2)/2

  =  (5/2, -3/2)

(i)  The required line is passing through the point (5/2, -3/2) and parallel to x axis.

If the line is parallel to x-axis, then slope of the required line = 0.

(y - y₁)  =  m (x - x₁)

y + (3/2)  =  0(x - (5/2))

(2y + 3) /2  =  0

2y + 3  =  0

So, the required line parallel to x-axis is

2y + 3  =  0

(ii)  The required line is passing through the point (5/2, -3/2) and parallel to x axis.

If the line is parallel to y-axis, then slope of the required line = undefined.

(y - y₁)  =  m (x - x₁)

y + (3/2)  =  (1/0)(x - (5/2))

x - (5/2)  =  0

2x - 5  =  0

So, the required line parallel to y-axis is

2x - 5  =  0

Example 2 :

The equation of a straight line is 2(x −y)+ 5 = 0 . Find its slope, inclination and intercept on the Y axis.

Solution :

2(x - y) + 5  =  0

2x - 2y + 5  =  0

2y  =  2x + 5

y  =  (2/2)x + (5/2)

y  = x + (5/2)

Slope (m)  =  1

Angle of inclination :

m = 1

tan θ  =  1

θ  =  45⁰

Intercept of y - axis :

y-intercept (c)  =  5/2

Example 3 :

Find the equation of a line whose inclination is 30˚ and making an intercept -3 on the Y axis.

Solution :

θ  =  30˚

m = tanθ

m = tan 30  =  1/√3

intercept on y - axis  =  -3

Equation of the line :

y = m x + c

y = (1/√3)x + (-3)

y = (x/√3) - (3√3)

√3y  =  x - 3√3

x - √3y - 3√3  =  0

Example 4 :

Find the slope and y intercept of √3x + (1 − √3)y = 3

Solution :

√3x +(1 − √3)y = 3

By comparing the given equation with the form y = mx + c, we get slope and y-intercept.

(1 − √3)y = 3 - √3x

y = (-√3x + 3)/(1 − √3)

y  = [-√3/(1 − √3)] x + 3/(1 − √3)

Slope   =  -√3/(1 − √3) 

  =  [ √3/(√3-1) ] (√3+1)/(√3+1)

  = √3(√3+1)/(√3-1)(√3+1)

  =  (3 + √3)/2

y-intercept   =  3/(1 − √3)

  =  [3/(1 − √3)][(1 + √3)/(1+√3)]

  =  3(1+√3)/(1 - 3)

  =  (-3 - 3√3)/2 

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