Coordinate SystemQuadrantsPlotting PointsDistance FormulaLinear GraphsKey PointsExercise 1Exercise 2
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Theory
Co-ordinate System
In two-dimensional coordinate geometry, we generally use two types of co-ordinate systems:
- (i) Cartesian or Rectangular Co-ordinate System — A point is represented by an ordered pair (x, y), where x is the x-coordinate and y is the y-coordinate.
- (ii) Polar Co-ordinate System — A point is represented by an ordered pair (r, θ), where r is the radius vector (distance from origin) and θ (theta) is the vectorial angle measured from the positive x-axis.
Ordered Pair
⚠️ Note: (p, q) and (q, p) represent two different points unless p = q.
Coordinate Axes
Draw two straight lines XOX′ and YOY′ intersecting at right angles at point O. These lines are called coordinate axes or axes of reference.
- The horizontal line XOX′ is the x-axis.
- The vertical line YOY′ is the y-axis.
- The point O where both axes meet is called the origin. Its coordinates are (0, 0).
- The distance measured along the x-axis is called the x-coordinate or abscissa.
- The distance measured along the y-axis is called the y-coordinate or ordinate.
- A point P(x, y): x is written first, y is written second.
Theory
Quadrants & Convention of Signs
The two coordinate axes XOX′ and YOY′ divide the plane into four regions called quadrants:
Both coordinates positive
x negative, y positive
Both coordinates negative
x positive, y negative
Points on the Axes
On the y-axis: The x-coordinate is zero. Points are of the form (0, y).
At the origin: Both coordinates are zero → (0, 0).
Q. In which quadrant do the given points lie?
(i) A(3, 4) (ii) B(−2, 3) (iii) C(−5, −2) (iv) D(4, −3) (v) E(−5, −5)
| Point | Signs (x, y) | Quadrant |
|---|---|---|
| A(3, 4) | (+, +) | I |
| B(−2, 3) | (−, +) | II |
| C(−5, −2) | (−, −) | III |
| D(4, −3) | (+, −) | IV |
| E(−5, −5) | (−, −) | III |
Q. Which of the following points lie on the x-axis or y-axis?
(−3, 4), (0, 8), (2, −3), (−6, 0)
(0, 8) → x-coordinate = 0 → lies on the y-axis.
(−6, 0) → y-coordinate = 0 → lies on the x-axis.
(−3, 4) and (2, −3) lie in Quadrant II and Quadrant IV respectively — neither axis.
Theory
Plotting of Points
To plot a point P(a, b) in the Cartesian plane, follow this algorithm:
Q. Plot the point (3, 4) on a graph paper.
Let X′OX and Y′OY be the coordinate axes. For P(3, 4):
Q. Plot the following points: A(2, 5), B(−5, −7), C(3, −2), D(0, 5), E(5, 0).
Formula
Distance Between Two Points
If there are two points A(x₁, y₁) and B(x₂, y₂) on the XY plane, the distance d between them is:
This is derived from the Pythagorean theorem: the horizontal and vertical separations form the two legs of a right triangle, and AB is the hypotenuse.
• Distance from origin to point P(x, y):
OP = √(x² + y²)• If two points lie on the x-axis (y = 0): distance = |x₂ − x₁|
• If two points lie on the y-axis (x = 0): distance = |y₂ − y₁|
Q. Find the distance between:
(i) (5, 3) and (3, 2) (ii) (−1, 4) and (2, −3) (iii) (a, b) and (−b, a)
(i) d₁ = √[(3 − 5)² + (2 − 3)²]
(ii) d₂ = √[(2 − (−1))² + (−3 − 4)²]
(iii) d₃ = √[(−b − a)² + (a − b)²]
Theory
Graph of Linear Equation ax + by + c = 0
The general linear equation ax + by + c = 0 can be rewritten as:
where:
- m = −a/b is called the slope (or gradient) of the line.
- c = −c₁/b is called the y-intercept (where the line cuts the y-axis).
Steps to Draw the Graph
Q. Draw the graph of the line y = mx + c.
Prepare a table of values. For x = 0, y = c. For y = 0, x = −c/m. For x = −c/m, we get x = −c/m.
| x | 0 | −c/m | −2c/m |
|---|---|---|---|
| y | c | 0 | −c |
1. The graph changes if the signs of m and c differ.
2. m (coefficient of x) is the slope of the line.
3. c is the y-intercept (where the line crosses the y-axis), i.e., OA = c.
Special Linear Graphs
Slope = 0
Slope = undefined
y-intercept = 0
Passes through origin
| Value of m | Position / Rotation | Quadrants |
|---|---|---|
| m > 0 (positive, increasing) | Line rotates anticlockwise as m increases | I and III |
| m < 0 (negative, decreasing) | Line rotates clockwise as |m| increases | II and IV |
| m = 0 | Horizontal line (y = 0, the x-axis) | — |
- Changing c to c₁, c₂, c₃ … gives parallel lines (same slope, different y-intercepts).
- Increasing c shifts the graph upward.
- Decreasing c shifts the graph downward.
The positioning of the graph depends on the sign combination of m and c:
| m | c | Graph Description |
|---|---|---|
| + | + | Rises left to right; crosses y-axis above origin |
| + | − | Rises left to right; crosses y-axis below origin |
| − | + | Falls left to right; crosses y-axis above origin |
| − | − | Falls left to right; crosses y-axis below origin |
Q. Draw the graph of y = 3x + 6. Find (i) slope (ii) y-intercept (iii) area between the line and axes.
Comparing with y = mx + c: m = 3, c = 6.
| x | 0 | −2 | −1 |
|---|---|---|---|
| y | 6 | 0 | 3 |
The line cuts the y-axis at (0, 6) and the x-axis at (−2, 0).
Summary
Key Points to Remember
- Two perpendicular lines XOX′ and YOY′ intersecting at O are thecoordinate axes.
- Origin O has coordinates(0, 0). The abscissa of every point on the y-axis is zero; the ordinate of every point on the x-axis is zero.
- The four quadrants have signs: I → (+,+), II → (−,+), III → (−,−), IV → (+,−).
- Theabscissa (x)is the perpendicular distance from the y-axis. Theordinate (y)is the perpendicular distance from the x-axis.
- Distance formula:
AB = √[(x₂−x₁)² + (y₂−y₁)²] - Iny = mx + c: m is the slope; c is the y-intercept.
- y = c→ parallel to x-axis; x = c→ parallel to y-axis.
- For y = mx with m > 0: graph is in Quadrants I & III and rotates anticlockwise as m increases.
- For y = mx with m < 0: graph is in Quadrants II & IV and rotates clockwise as |m| increases.
- Changing c shifts parallel lines: increasing c moves graph up; decreasing c moves graph down.
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Practice
Let's Try — Quick Practice
| # | Question |
|---|---|
| 1 | In which quadrant do the given points lie? (i) (3, −5) (ii) (−4, 6) (iii) (−3, −1) (iv) (2, 5) |
| 2 | Which of the given points lie on the x or y axis? (−3, 4), (0, 8), (2, −3), (−6, 0) |
| 3 | Plot the points O(0,0), A(16,0), B(16,12) on graph paper. Join OB, OA and BA. Name the figure and find its area. |
| 4 | Draw the graph of y = 3x + 6. Find (i) slope (ii) intercept on y-axis (iii) area between the line and axes. |
1. (i) IV quadrant (ii) II quadrant (iii) III quadrant (iv) I quadrant
2. (0, 8) → y-axis; (−6, 0) → x-axis
3. Right triangle; Area = ½ × 16 × 12 = 96 square units
4. Slope = 3; y-intercept OA = 6 units; Area = 6 square units
Exercise 1
MCQ Exercise – 1
- 1. Which point lies on the x-axis?(a) (3, 2)(b) (−3, 2)(c) (2, 0) ✓(d) (1, 2)
- 2. Which point lies on the y-axis?(a) (1, 3)(b) (0, 3) ✓(c) (5, 2)(d) (2, 3)
- 3. Which point lies above the x-axis?(a) (1, 2) ✓(b) (2, 0)(c) (1, −5)(d) (0, −3)
- 4. Which point lies to the left of the y-axis?(a) (2, 0)(b) (−3, 4) ✓(c) (4, 2)(d) (2, 7)
- 5. Which point lies to the right of the y-axis?(a) (0, 3)(b) (−2, 1)(c) (2, −5) ✓(d) (−3, 2)
- 6. Which line is parallel to y = x − 2?(a) y = 2x + 1(b) 2y = 2x − 6 ✓(c) 2y = x + 7(d) y = 3x + 1
Reason: 2y = 2x − 6 simplifies to y = x − 3, which has slope 1 — same as y = x − 2. Parallel lines have equal slopes.
- 7. Which point lies in the IV quadrant?(a) (2, 3)(b) (1, −4) ✓(c) (−2, 3)(d) (0, 1)
- 8. Which point lies in the III quadrant?(a) (0, −3)(b) (2, 0)(c) (7, −2)(d) (−2, −3) ✓
- 9. Which graph is parallel to the x-axis?(a) y = x + 1(b) y = 2 ✓(c) x = 3(d) x = 2y
- 10. Which graph is equally inclined to both axes?(a) y = 2(b) x = 2y(c) 2y = x(d) 2y = 2x ✓ (i.e., y = x)
- 11. The abscissa of a point is the distance of the point from:(a) X-axis(b) Y-axis ✓(c) Origin(d) None of these
- 12. The y-coordinate of a point is the distance of that point from:(a) X-axis ✓(b) Y-axis(c) Origin(d) None of these
- 13. If both coordinates of any point are negative, then that point will lie in:(a) First quadrant(b) Second quadrant(c) Third quadrant ✓(d) Fourth quadrant
- 14. If the abscissa of any point is zero, then that point will lie:(a) on X-axis(b) on Y-axis ✓(c) at origin(d) None of these
- 15. One end of a diameter of a circle is (4, −1) and the centre is (1, −3). The coordinates of the other end are:(a) (2, 5)(b) (−2, −5) ✓(c) (3, 2)(d) (−3, −2)
Using mid-point formula: Centre = midpoint → (1, −3) = ((4+x)/2, (−1+y)/2) → x = −2, y = −5.
- 16. The points (−2,−1), (1,0), (4,3) and (1,2) are the vertices of a:(a) Rectangle(b) Parallelogram ✓(c) Square(d) Rhombus
- 17. The distance of the point (3, 5) from the x-axis is:(a) √34(b) 3(c) 5 ✓(d) None of these
The distance from the x-axis is simply the ordinate (y-coordinate) = 5.
1.(c) · 2.(b) · 3.(a) · 4.(b) · 5.(c) · 6.(b) · 7.(b) · 8.(d) · 9.(b) · 10.(d) · 11.(b) · 12.(a) · 13.(c) · 14.(b) · 15.(b) · 16.(b) · 17.(c)
Exercise 2
Descriptive Exercise – 2
| # | Question | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | Plot the points in the plane if coordinates are: A(5,0), B(0,3), C(7,2), D(−4,3), E(−3,−2) and F(3,−2). | ||||||||||||
| 2 | In which quadrant do the following points lie? A(2,3), B(−2,3), C(−3,−5), D(3,−1). Explain with reasons. | ||||||||||||
| 3 | Plot the following pairs of numbers as points in the Cartesian plane:
| ||||||||||||
| 4 | With rectangular axes, plot O(0,0), A(4,0) and C(0,6). Find the coordinates of the fourth point B such that OABC forms a rectangle. | ||||||||||||
| 5 | Plot P(−3,1) and Q(2,1) in the rectangular coordinate system. Find all possible coordinates of the other two vertices of a square having P and Q as two adjacent vertices. | ||||||||||||
| 6 | Find the value of x if the distance between (x, −1) and (3, 2) is 5. | ||||||||||||
| 7 | The base AB of two equilateral triangles ABC and ABC′ with side 2a lies along the x-axis such that the midpoint of AB is at the origin. Find the coordinates of C and C′. | ||||||||||||
| 8 | Name the quadrants in which the following points lie: (a) (2, 3) (b) (3, −2). | ||||||||||||
| 9 | Determine the slope and y-intercept of the graph of y = (1/2)x − 2. | ||||||||||||
| 10 | Draw the graph of the equation y = 2x + 3. | ||||||||||||
| 11 | Draw the graph of y = 4x. From the graph find the value of y when x = 2. | ||||||||||||
| 15 | Draw the graph of 3x + 6y = 12. Find the coordinates of the point where the graph cuts the y-axis. | ||||||||||||
| 16 | Draw the graph of x + 10 = 0. What type of graph is it? | ||||||||||||
| 17 | The points (2, 5) and (3, 5) are plotted in xy plane. Find the slope and y-intercept of the line joining the points. [Hint: both points have same y-coordinate] | ||||||||||||
| 18 | Draw the graphs of (i) y = x, (ii) y = x + 1, (iii) y = x + 2 on the same axes. | ||||||||||||
| 19 | Draw the graphs of (i) y = 2x, (ii) y = 2x + 1, (iii) y = 2x + 3 on the same axes. | ||||||||||||
| 20 | Draw the graphs of (i) y = 2x, (ii) y = 3x, (iii) y = 4x on the same axes. |
2. A → I, B → II, C → III, D → IV
4. B = (4, 6)
5. Other vertices: (−3, 6) & (2, 6) or (−3, −4) & (2, −4)
6. x = 7 or x = −1
7. C = (0, a√3), C′ = (0, −a√3)
8. (a) Quadrant I (b) Quadrant IV
9. Slope = 1/2, y-intercept = −2
11. When x = 2, y = 8 (from y = 4x)
15. Graph cuts y-axis at (0, 2)
16. x = −10 is a vertical line parallel to the y-axis
17. Slope = 0 (horizontal line), y-intercept = 5
Notes
Make Your Own Notes
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FAQs on CBSE Class 9 Maths Notes Coordinate Geometry – Concepts & FAQs
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