To understand the equation of a line which is parallel to x axis,
To understand the equation of a line which is parallel to 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 = (x_{1} + x_{2})/2, (y_{1} + y_{2})/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 - y_{1}) = m (x - x_{1})
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 - y_{1}) = m (x - x_{1})
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 = 45^{0}
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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