How to Find the General Equation of a Line Given Two Points
The concept of general equation of a line plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It helps in representing all types of straight lines in a plane and is essential for board exams, JEE, and Olympiad preparation. Mastering the general equation of a line allows students to solve geometry, algebra, and coordinate geometry problems quickly and with confidence.
What Is General Equation of a Line?
A general equation of a line is a way to represent any straight line on a coordinate plane using the formula Ax + By + C = 0. Here, A and B cannot both be zero, and A, B, and C are constants. You’ll find this concept applied in areas such as geometry (lines and angles), algebraic linear equations, and coordinate geometry topics like parallel and perpendicular lines.
Key Formula for General Equation of a Line
Here’s the standard formula: \( Ax + By + C = 0 \)
Where:
x, y = variables representing coordinates (points) on the line.
If you know two points the line passes through, or its slope and a point, you can substitute into Ax + By + C = 0 to find the line’s equation.
Cross-Disciplinary Usage
The general equation of a line is not only useful in Maths but also plays an important role in Physics (motion, optics), Computer Science (graphics, algorithms), and daily logical reasoning. Students preparing for JEE, NEET, and various board exams will see its relevance in multiple questions involving coordinate geometry, graph plotting, and analytical geometry.
Step-by-Step Illustration
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Suppose you have two points: (2, 3) and (4, 7). Find the general equation of the line through these points.
Step 1: Calculate the slope (m):
\( m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \)
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Step 2: Use point-slope form:
\( y - y_1 = m(x - x_1) \)
Pick (2, 3): \( y - 3 = 2(x - 2) \)
So, \( y - 3 = 2x - 4 \) -
Step 3: Rearrange to general form:
\( y - 3 - 2x + 4 = 0 \)
So, \( -2x + y + 1 = 0 \)
Or, \( 2x - y - 1 = 0 \)
Special Cases for General Equation of a Line
| Case | Equation | Description |
|---|---|---|
| Vertical Line | x = a | B = 0, line parallel to y-axis |
| Horizontal Line | y = b | A = 0, line parallel to x-axis |
| Through Origin | Ax + By = 0 | C = 0 |
| General 3D Line | Parametric equations (not Ax + By + C = 0) | For 3D, two equations required |
Comparison With Other Forms of a Line
| Form | Equation | Conversion to General |
|---|---|---|
| Slope-Intercept | \( y = mx + c \) | Rearrange to \( mx - y + c = 0 \) |
| Point-Slope | \( y - y_1 = m(x - x_1) \) | Expand, rearrange terms |
| Intercept | \( x/a + y/b = 1 \) | Multiply by denominator, bring all to one side |
Speed Trick or Revision Shortcut
When given two points (x₁, y₁) and (x₂, y₂), use this quick method:
- Use determinant formula:
\( (y - y_1)(x_2 - x_1) = (y_2 - y_1)(x - x_1) \) - Expand and rearrange terms to get \( Ax + By + C = 0 \).
- This method is fast and avoids direct slope calculation errors during exams.
Vedantu’s teachers often share such tricks in live sessions to boost students’ accuracy and calculation speed.
Try These Yourself
- Find the general equation of a line passing through (1,2) and (3,6).
- Convert \( y = -2x + 5 \) to general form.
- Does \( 4x + 3y - 7 = 0 \) represent a line parallel to the y-axis?
- Write the general equation for a line with slope 5 that passes through (0,0).
Frequent Errors and Misunderstandings
- Forgetting to move all terms to one side, leaving the equation in slope-intercept or point-slope form by mistake.
- Mixing up the sign of coefficients when rearranging.
- Not checking if A, B, C are integers (as sometimes asked in exams).
- Setting both A and B to zero, which does not form a valid line.
Relation to Other Concepts
The idea of general equation of a line connects closely with other line forms, cartesian coordinates, and the calculation of distance from a point to a line. Mastering this helps with more advanced topics such as intersection of lines and equations in 3D geometry.
Classroom Tip
A quick way to remember the general equation of a line: Always bring all x and y terms to one side and set the equation to zero (\( Ax + By + C = 0 \)). If you ever get confused, check your units, and signs of coefficients. Vedantu’s tutors use visual aids to help students see how different forms of a line convert to the general form during class explanations.
We explored general equation of a line—from definition, formula, and key properties to tricky mistakes and its links to other key topics. Continue practicing with Vedantu and use their tips to feel confident when solving line equation problems in your exams and competitive tests!
Related Topics on Vedantu:
FAQs on General Equation of a Line: Formula, Derivation, and Applications
1. What is the general equation of a straight line?
The general equation of a straight line is Ax + By + C = 0, where A, B, and C are constants (real numbers), and x and y are variables representing the coordinates of points on the line. This form represents all possible straight lines except vertical lines (which have the form x = constant).
2. How do you find the general equation of a line given two points?
First, find the slope (m) using the formula: m = (y₂ - y₁) / (x₂ - x₁). Then, use the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is one of the points. Finally, rearrange this equation into the form Ax + By + C = 0.
3. What is the difference between the slope-intercept and general forms of a line equation?
The slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept. The general form is Ax + By + C = 0. The general form is more versatile for solving problems involving intersecting lines and distances, while the slope-intercept form is useful for quickly graphing a line.
4. How do you convert the slope-intercept form (y = mx + c) to the general form (Ax + By + C = 0)?
To convert, rearrange the equation so that all terms are on one side and equal to zero: mx - y + c = 0. This is already in the general form, where A = m, B = -1, and C = c.
5. Can the general equation describe all possible lines?
Yes, the general equation Ax + By + C = 0 describes all straight lines except vertical lines. Vertical lines have the equation x = k, where k is a constant.
6. What if A or B is zero in the general equation?
If A = 0, the line is horizontal (parallel to the x-axis) and its equation simplifies to By + C = 0 or y = -C/B. If B = 0, the line is vertical (parallel to the y-axis) and its equation becomes Ax + C = 0 or x = -C/A.
7. How do I find the x- and y-intercepts from the general equation?
To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. These points are where the line crosses the x and y axes respectively.
8. How can I determine if two lines are parallel or perpendicular using their general equations?
For two lines in general form, A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0: They are parallel if A₁/A₂ = B₁/B₂. They are perpendicular if A₁A₂ + B₁B₂ = 0.
9. What are some common mistakes students make with the general equation of a line?
Common errors include incorrect calculation of the slope, sign errors during rearrangement, and forgetting to check the final equation with known points. Carefully verifying each step is essential.
10. How is the general equation of a line useful in solving geometric problems?
The general equation simplifies calculations for finding the intersection point of two lines, the distance between a point and a line, and determining whether lines are parallel or perpendicular. It provides a unified approach to various line-related problems.
11. What is the significance of the constants A, B, and C in Ax + By + C = 0?
The ratio A/B determines the slope of the line. The constant C affects the line's position on the coordinate plane; it determines the y-intercept when A is nonzero. The values of A and B, and their relative magnitudes, will influence the angle and position of the line.
