Given the linear equation 2x + 3y − 8 = 0, write another linear equation such that the geometrical representation of the pair is intersecting, parallel and coincident
Video Explanation
Watch the video explanation below:
Given
First linear equation:
2x + 3y − 8 = 0
To Write
Another linear equation in each case so that the pair represents:
- Intersecting lines
- Parallel lines
- Coincident lines
(i) Intersecting Lines
For intersecting lines:
a₁/a₂ ≠ b₁/b₂
Take another equation:
x + y − 5 = 0
Here,
a₁/a₂ = 2/1 = 2
b₁/b₂ = 3/1 = 3
Since a₁/a₂ ≠ b₁/b₂,
The lines intersect at a point.
(ii) Parallel Lines
For parallel lines:
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Take another equation:
4x + 6y − 10 = 0
Here,
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = 3/6 = 1/2
c₁/c₂ = −8/−10 = 4/5
Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂,
The lines are parallel.
(iii) Coincident Lines
For coincident lines:
a₁/a₂ = b₁/b₂ = c₁/c₂
Take another equation:
4x + 6y − 16 = 0
Here,
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = 3/6 = 1/2
c₁/c₂ = −8/−16 = 1/2
Since a₁/a₂ = b₁/b₂ = c₁/c₂,
The lines coincide.
Final Answer
- Intersecting lines: 2x + 3y − 8 = 0 and x + y − 5 = 0
- Parallel lines: 2x + 3y − 8 = 0 and 4x + 6y − 10 = 0
- Coincident lines: 2x + 3y − 8 = 0 and 4x + 6y − 16 = 0
Conclusion
Thus, by choosing suitable coefficients and comparing ratios, we can form pairs of linear equations representing intersecting, parallel and coincident lines.