Coordinate Geometry and Straight Line: Formulas, Slope and Equations of Line for SSC
Rahul Kumar
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Coordinate geometry joins algebra and geometry by describing points and lines with numbers. In SSC CGL, CPO and CHSL it gives easy and direct questions on distance, midpoint, centroid, slope and the equation of a line. If you remember about ten formulas, you can solve most of these in under 30 seconds. This note gives every formula with a solved example, a revision list, practice questions and FAQs.
Points and Quadrants
| Quadrant | Sign of (x, y) |
|---|---|
| I | (+, +) |
| II | (−, +) |
| III | (−, −) |
| IV | (+, −) |
- Any point on the x-axis has y = 0, i.e. (x, 0). Any point on the y-axis has x = 0, i.e. (0, y).
- The origin is (0, 0).
- The reflection of (a, b) in the x-axis is (a, −b); in the y-axis it is (−a, b); in the origin it is (−a, −b).
Important Formulas
| Concept | Formula |
|---|---|
| Distance between (x1, y1) and (x2, y2) | √[(x2 − x1)² + (y2 − y1)²] |
| Distance of (x, y) from the origin | √(x² + y²) |
| Midpoint | ((x1 + x2)/2, (y1 + y2)/2) |
| Section formula (internal, ratio m : n) | ((m·x2 + n·x1)/(m + n), (m·y2 + n·y1)/(m + n)) |
| Centroid of a triangle | ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3) |
| Area of a triangle | ½ |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)| |
| Three points are collinear | The area of the triangle formed = 0 |
Slope of a Line
- Slope m = (y2 − y1)/(x2 − x1) = tan θ, where θ is the angle made with the positive x-axis.
- Slope of a horizontal line (parallel to the x-axis) = 0; a vertical line has undefined slope.
- Parallel lines have equal slopes: m1 = m2.
- Perpendicular lines have m1 × m2 = −1.
- For ax + by + c = 0, slope = −a/b.
- Angle between two lines: tan θ = |(m1 − m2)/(1 + m1·m2)|.
Equations of a Straight Line
| Form | Equation |
|---|---|
| Slope-intercept | y = mx + c (c is the y-intercept) |
| Point-slope | y − y1 = m(x − x1) |
| Two-point | (y − y1)/(y2 − y1) = (x − x1)/(x2 − x1) |
| Intercept form | x/a + y/b = 1 (a = x-intercept, b = y-intercept) |
| General form | ax + by + c = 0 |
| Parallel to the x-axis / y-axis | y = k / x = k |
Distance Formulas for a Line
- Distance of the point (x1, y1) from the line ax + by + c = 0: |ax1 + by1 + c| / √(a² + b²).
- Distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0: |c1 − c2| / √(a² + b²).
- Equation of a circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r².
Solved Examples
| Question | Solution |
|---|---|
| Distance between (3, 4) and (7, 1) | √(16 + 9) = √25 = 5 |
| Midpoint of (2, −3) and (8, 5) | ((2 + 8)/2, (−3 + 5)/2) = (5, 1) |
| Point dividing (1, 2) and (7, 8) in the ratio 1 : 2 | ((1×7 + 2×1)/3, (1×8 + 2×2)/3) = (3, 4) |
| Centroid of (0, 0), (6, 0), (0, 9) | (6/3, 9/3) = (2, 3) |
| Slope of the line through (1, 2) and (3, 8) | (8 − 2)/(3 − 1) = 3 |
| Line with slope 2 through (1, 3) | y − 3 = 2(x − 1) ⇒ y = 2x + 1 |
| Distance of (3, 4) from 3x + 4y − 5 = 0 | |9 + 16 − 5|/5 = 20/5 = 4 |
| Intercepts of 2x + 3y = 12 | x-intercept 6, y-intercept 4; area with the axes = ½ × 6 × 4 = 12 |
| Slope of a line perpendicular to 3x − y + 2 = 0 | Slope of the given line = 3, so required slope = −1/3 |
| Are (1, 1), (2, 3), (3, 5) collinear? | 1(3 − 5) + 2(5 − 1) + 3(1 − 3) = −2 + 8 − 6 = 0, so yes |
Common Mistakes
- Forgetting to take the modulus (absolute value) in the area and point-to-line distance formulas.
- In the section formula, giving n to x1 and m to x2. The ratio m : n is measured from the first point.
- Using the slope formula with the points in different orders for the numerator and the denominator.
- Taking the slope of ax + by + c = 0 as a/b instead of −a/b.
Quick Tips
- Write the formula first and substitute carefully; do not do it in your head.
- Use slope to check parallel and perpendicular lines in a few seconds.
- For the area of a triangle with a vertex at the origin, the formula reduces to ½|x1y2 − x2y1|.
- Check the answer with a rough sketch when the numbers are small.
Quick Revision Points
- Distance = √[(x2 − x1)² + (y2 − y1)²].
- Midpoint = average of the x values and of the y values.
- Slope = (y2 − y1)/(x2 − x1); parallel m1 = m2; perpendicular m1m2 = −1.
- Point to line distance = |ax1 + by1 + c|/√(a² + b²).
- Collinear points give zero area.
Practice Questions With Answers
- Distance of (6, 8) from the origin?
Answer: 10 - Midpoint of (4, 6) and (10, 2)?
Answer: (7, 4) - Slope of the line 2x + y = 7?
Answer: −2 - x-intercept of 3x + 4y = 12?
Answer: 4 - Centroid of (3, 6), (9, 0), (0, 3)?
Answer: (4, 3) - Area of the triangle with vertices (0, 0), (6, 0), (0, 8)?
Answer: 24 square units - Distance of the origin from 3x + 4y = 10?
Answer: 2 - Midpoint of (2, 3) and (8, 9)?
Answer: (5, 6) - Are y = 3x + 2 and y = 3x − 5 parallel?
Answer: Yes, both have slope 3 - Slope of a line perpendicular to a line of slope 1/2?
Answer: −2
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Coordinate Geometry and Straight Line notes page.
Frequently Asked Questions
What is coordinate geometry?
It is the study of geometry using a coordinate system, where every point is described by an ordered pair (x, y).
What is the distance formula?
The distance between (x1, y1) and (x2, y2) is the square root of (x2 − x1)² + (y2 − y1)².
How do I find the midpoint of a line segment?
Take the average of the x coordinates and the average of the y coordinates.
What is the section formula?
It gives the point that divides a line joining two points in the ratio m : n.
How do I find the slope of a line?
Divide the change in y by the change in x between any two points on the line.
What is the condition for two lines to be perpendicular?
The product of their slopes is −1.
What is the slope of the line ax + by + c = 0?
It is −a/b.
How do I check if three points are collinear?
Find the area of the triangle formed by them. If it is zero, the points are collinear.
How many questions come from coordinate geometry in SSC?
Usually 1 to 2 questions in CGL Tier 1 and a few in Tier 2, depending on the paper.
What is the equation of a line parallel to the x-axis?
y = k, where k is a constant.
What is the centroid of a triangle?
The point where the three medians meet, at the average of the three vertices.
What is the equation of a circle with centre (h, k)?
(x − h)² + (y − k)² = r².
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Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.