Coordinate Geometry Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
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PDF Notes: Coordinate Geometry Notes for SSC Exams
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Coordinate geometry uses numbers (coordinates) to describe points, lines and shapes on a plane. In SSC exams it is a scoring topic, because most questions use a few formulas: distance, midpoint, centroid, slope and the area of a triangle. These coordinate geometry notes for SSC explain every formula with solved examples and practice questions in simple language.
Basics of the Coordinate Plane
- The plane has the x-axis (horizontal) and the y-axis (vertical), which meet at the origin (0, 0).
- A point is written as (x, y), where x is the distance along the x-axis and y is the distance along the y-axis.
- The plane is divided into four quadrants.
| Quadrant | Sign of x | Sign of y |
|---|---|---|
| First | Positive | Positive |
| Second | Negative | Positive |
| Third | Negative | Negative |
| Fourth | Positive | Negative |
- The equation of the x-axis is y = 0, and the equation of the y-axis is x = 0.
- Any point on the x-axis has the form (a, 0), and any point on the y-axis has the form (0, b).
Important Formulas
| Item | Formula for points A(x1, y1) and B(x2, y2) |
|---|---|
| Distance between two points | AB = √[(x2 − x1)2 + (y2 − y1)2] |
| Distance from the origin | √(x2 + y2) |
| Midpoint of AB | ((x1 + x2) / 2, (y1 + y2) / 2) |
| Internal section formula (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 | (1/2) |x1 (y2 − y3) + x2 (y3 − y1) + x3 (y1 − y2)| |
| Three points are collinear | Area of the triangle formed by them is zero |
Solved Examples
Example 1. Find the distance between (1, 2) and (4, 6).
Distance = √(32 + 42) = √25 = 5.
Example 2. Find the midpoint of (2, 3) and (6, 9).
Midpoint = (8/2, 12/2) = (4, 6).
Example 3. Find the point that divides (1, 2) and (7, 8) in the ratio 1 : 2.
x = (1 × 7 + 2 × 1) / 3 = 3 and y = (1 × 8 + 2 × 2) / 3 = 4, so the point is (3, 4).
Example 4. Find the centroid of the triangle with vertices (0, 0), (6, 0) and (0, 9).
Centroid = (6/3, 9/3) = (2, 3).
Example 5. Find the area of the triangle with vertices (0, 0), (4, 0) and (0, 3).
Area = (1/2) |0 + 4 (3 − 0) + 0| = 6 square units.
Slope and Equation of a Line
| Item | Formula |
|---|---|
| Slope (gradient) | m = (y2 − y1) / (x2 − x1) = tan θ |
| Slope-intercept form | y = m x + c |
| Point-slope form | y − y1 = m (x − x1) |
| Two-point form | (y − y1) / (y2 − y1) = (x − x1) / (x2 − x1) |
| Intercept form | x / a + y / b = 1 |
| General form | A x + B y + C = 0, with slope = −A / B |
- Parallel lines have equal slopes (m1 = m2).
- Perpendicular lines have slopes whose product is −1 (m1 × m2 = −1).
- The slope of a horizontal line is 0, and a vertical line has no slope (undefined).
- Distance of a point (x1, y1) from the line A x + B y + C = 0 is |A x1 + B y1 + C| / √(A2 + B2).
Example 6. Find the slope of the line through (1, 2) and (3, 6).
m = (6 − 2) / (3 − 1) = 2.
Example 7. Find the distance of the origin from the line 3x + 4y − 10 = 0.
Distance = |−10| / √(9 + 16) = 10 / 5 = 2.
Equation of a Circle
- A circle with centre (h, k) and radius r has the equation (x − h)2 + (y − k)2 = r2.
- If the centre is the origin, the equation is x2 + y2 = r2.
Quick Tips
- Write the formula first and then put the coordinates, taking care of the signs.
- Use the midpoint and the distance formulas to check the type of triangle or quadrilateral.
- For collinear points, check that the area is zero or that the slopes are equal.
- Draw a quick sketch of the points before solving.
Quick Revision Points
- Distance = √[(x2 − x1)2 + (y2 − y1)2].
- Midpoint = average of the x and y coordinates.
- Centroid = average of the three vertices.
- Slope m = (y2 − y1) / (x2 − x1).
- Parallel lines have equal slopes and perpendicular lines have m1 m2 = −1.
Practice Questions With Answers
- Find the distance between (0, 0) and (6, 8).
Answer: 10 - Find the midpoint of (4, 2) and (10, 8).
Answer: (7, 5) - Find the centroid of (1, 2), (3, 4) and (5, 9).
Answer: (3, 5) - Find the slope of the line through (2, 3) and (4, 7).
Answer: 2 - Find the area of the triangle with vertices (0, 0), (6, 0) and (0, 8).
Answer: 24 square units - What is the equation of the x-axis?
Answer: y = 0 - What is the slope of a line parallel to y = 3x + 5?
Answer: 3 - If two lines are perpendicular and one has slope 2, what is the slope of the other?
Answer: −1/2 - Find the radius of the circle x2 + y2 = 49.
Answer: 7 - In which quadrant does the point (−3, 4) lie?
Answer: Second quadrant
How to Practise This Chapter
Solve five coordinate geometry questions daily, with one question on each formula, and draw a sketch each time. Then attempt timed sets from our practice papers and the previous year papers page. Read the next chapter, Arithmetic and Geometric Progression, and see all topics on the Mathematics notes page. For more maths, read the Quantitative Aptitude notes.
Frequently Asked Questions
What is coordinate geometry?
Coordinate geometry is the study of points, lines and shapes using their coordinates on a plane.
What is the distance formula?
The distance between two points is the square root of the sum of the squares of the differences of their x and y coordinates.
What is the midpoint formula?
The midpoint of two points has coordinates equal to the average of their x values and the average of their y values.
What is the section formula?
It gives the coordinates of the point that divides the line joining two points in a given ratio m to n.
How do I find the centroid of a triangle?
Add the x coordinates of the three vertices and divide by 3, and do the same for the y coordinates.
How do I find the area of a triangle using coordinates?
Use half of the absolute value of x1 times (y2 minus y3) plus x2 times (y3 minus y1) plus x3 times (y1 minus y2).
What is the slope of a line?
Slope is the change in y divided by the change in x between two points of the line, and it equals tan of the angle with the x-axis.
What is the condition for two lines to be parallel or perpendicular?
Parallel lines have equal slopes, and for perpendicular lines the product of the slopes is minus 1.
What is the equation of a circle?
A circle with centre (h, k) and radius r has the equation (x minus h) squared plus (y minus k) squared equals r squared.
How do I check if three points are collinear?
The three points are collinear if the area of the triangle formed by them is zero.
Is coordinate geometry important for SSC?
Yes. It gives easy formula-based questions in SSC CGL and CHSL, especially in the higher tiers.
How many questions come from coordinate geometry in SSC exams?
Usually one or two questions come from coordinate geometry, but the number changes from year to year.
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.