Coordinate Geometry

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CLASS IX Mathematics ~4 marks/year Ch 3 of 12
Coordinate Geometry

Class 9 · Mathematics · NCERT chapter notes · Akanksha Classes

Snapshot
  • Coordinate Geometry is the bridge between algebra and geometry — it lets you give every point on a flat plane an exact numerical address.
  • Two perpendicular number lines (the x-axis and y-axis) create the Cartesian plane; they cross at the origin O = (0, 0).
  • The address of any point is an ordered pair (x, y): the abscissa (x) tells how far right/left from the y-axis; the ordinate (y) tells how far up/down from the x-axis.
  • The axes divide the plane into four quadrants (I, II, III, IV), numbered anti-clockwise from the top-right, with fixed sign patterns.
  • Points on the x-axis have $y = 0$; points on the y-axis have $x = 0$; they belong to no quadrant.
  • Board weightage: ~4 marks/year — typically one 1-mark MCQ (quadrant/axis) and one 3-mark plotting question.
Detailed Notes

1. Historical Introduction — René Descartes

Coordinate Geometry was invented by the French mathematician René Descartes (1596–1650 CE) and is therefore called the Cartesian system in his honour. The story goes that while lying in bed one morning, Descartes watched a fly crawling on his ceiling and wondered: can I describe exactly where the fly is using numbers? He realised that if he chose two fixed lines on the ceiling meeting at right angles, the fly's position could be pinned down by measuring its distance from each line. That single insight fused algebra and geometry into one powerful discipline.

Before Descartes, algebra and geometry were entirely separate subjects. His invention meant that geometric shapes could be expressed as equations, and equations could be drawn as shapes — opening the door to calculus, physics, engineering, and computer graphics.

Why it matters for Class 9: you will use the Cartesian system constantly — in graphing linear equations (Chapter 4), in statistics (Chapter 14), and in higher classes for circles, parabolas, and beyond. Mastering coordinates now pays dividends for years.

2. The Cartesian Plane — Axes, Origin, and Quadrants

Start with a horizontal number line and call it the x-axis (written X'OX). Draw a vertical number line through the same zero point so that it is perpendicular (at 90°) to the x-axis; call it the y-axis (written Y'OY). Together these two lines are the coordinate axes. The flat surface on which they lie is the Cartesian plane, also called the coordinate plane or the xy-plane.

The point where the axes cross is called the origin, labelled O. Its coordinates are (0, 0) — it is zero distance from both axes.

Direction convention on each axis:

  • x-axis: numbers increase to the right of O (positive side, OX) and decrease to the left (negative side, OX').
  • y-axis: numbers increase going upward from O (positive side, OY) and decrease going downward (negative side, OY').

The two axes divide the entire plane into four regions called quadrants, numbered in anti-clockwise order starting from the top-right:

  • Quadrant I — between OX and OY (top-right region).
  • Quadrant II — between OX' and OY (top-left region).
  • Quadrant III — between OX' and OY' (bottom-left region).
  • Quadrant IV — between OX and OY' (bottom-right region).

3. Coordinates of a Point — Abscissa and Ordinate

To pinpoint any location P in the plane, we need two numbers. From point P, draw a perpendicular onto the x-axis; the value where it meets the x-axis is the x-coordinate (also called the abscissa). Draw another perpendicular from P onto the y-axis; the value where that meets the y-axis is the y-coordinate (also called the ordinate).

We write them together as the ordered pair (x, y). The word "ordered" is crucial: (3, 5) and (5, 3) are two completely different points, because the x-coordinate always comes first and the y-coordinate always comes second.

$$P = (x,\; y) = (\text{abscissa},\; \text{ordinate})$$ $$\text{abscissa} = \text{distance of P from the y-axis (with sign)}$$ $$\text{ordinate} = \text{distance of P from the x-axis (with sign)}$$

Memory trick: in the alphabet, x comes before y — so in the bracket, x comes before y. And "run before you climb": first move sideways (x), then climb up or down (y).

4. Quadrant Sign Convention Table

The sign of x tells you whether the point is to the right (+) or left (–) of the y-axis. The sign of y tells you whether it is above (+) or below (–) the x-axis. Together, the signs determine the quadrant:

Quadrant Sign of x Sign of y Region Example point
I + (positive) + (positive) Top-right (3, 5)
II – (negative) + (positive) Top-left (–4, 7)
III – (negative) – (negative) Bottom-left (–2, –6)
IV + (positive) – (negative) Bottom-right (5, –3)

Anti-clockwise trick: imagine writing the letter C — you start top-right (I), go top-left (II), bottom-left (III), finish bottom-right (IV). That anti-clockwise sweep is the standard numbering.

5. Points on the Axes

Points do not have to be inside a quadrant — some lie exactly on an axis. The rules are simple:

  • If a point lies on the x-axis, it has zero distance from the x-axis, so its y-coordinate is 0. It is written as $(x,\; 0)$. Examples: $(6,\; 0)$, $(-2,\; 0)$, $(0,\; 0)$.
  • If a point lies on the y-axis, it has zero distance from the y-axis, so its x-coordinate is 0. It is written as $(0,\; y)$. Examples: $(0,\; 4)$, $(0,\; -3)$.
  • The origin O = (0, 0) lies on both axes simultaneously.
  • Key rule: points on the axes do not belong to any quadrant. Quadrants are the open regions between the axes, not the axes themselves. This is a very common exam trap!

6. Plotting Points — Step-by-Step Method

To plot a point P(a, b) on the Cartesian plane:

  1. Draw the x-axis and y-axis with a ruler. Mark equal units (choose a scale, e.g., 1 cm = 1 unit) on both axes. Label the axes and the origin O.
  2. Look at the x-coordinate (abscissa) a. If positive, count |a| units to the right of O along the x-axis. If negative, count |a| units to the left. Mark a small tick on the x-axis.
  3. From that tick on the x-axis, move parallel to the y-axis: if y-coordinate (ordinate) b is positive, go up b units; if negative, go down |b| units.
  4. Mark a dot at the final position. Label it P(a, b).

Reading coordinates from a graph (reverse process): given a plotted point, draw a perpendicular from the point to the x-axis (read the x-coordinate) and another perpendicular to the y-axis (read the y-coordinate). Combine as (x, y).

Scale note: the scales on the two axes do not have to be the same. For example, you might use 1 cm = 2 units on the x-axis and 1 cm = 5 units on the y-axis if the y-values are much larger. Always state the scale used.

7. Distance of a Point from the Axes

This is a straightforward but frequently tested concept:

  • The distance of a point P(x, y) from the x-axis = $|y|$ (the absolute value of the ordinate).
  • The distance of a point P(x, y) from the y-axis = $|x|$ (the absolute value of the abscissa).

Think about why: the ordinate tells you how far the point is above or below the x-axis; the sign tells direction but distance is always non-negative. Similarly for the abscissa and y-axis.

Examples:

  • $P(3, -7)$: distance from x-axis $= |-7| = 7$ units; distance from y-axis $= |3| = 3$ units.
  • $Q(-5, 4)$: distance from x-axis $= |4| = 4$ units; distance from y-axis $= |-5| = 5$ units.
  • $R(-2, -9)$: distance from x-axis $= |-9| = 9$ units; distance from y-axis $= |-2| = 2$ units.

8. Additional Key Facts

  • Two points having the same x-coordinate lie on a vertical line parallel to the y-axis.
  • Two points having the same y-coordinate lie on a horizontal line parallel to the x-axis.
  • If a point $(a, b)$ is reflected in the x-axis, the image is $(a, -b)$. Reflected in the y-axis, image is $(-a, b)$. Reflected through the origin, image is $(-a, -b)$.
  • To check if three plotted points form a specific shape (triangle, rectangle, square), use the distance between pairs of plotted points — this connects to the Distance Formula covered in Class 10.
  • The point $(-x, y)$ is the mirror image of $(x, y)$ across the y-axis. The point $(x, -y)$ is the mirror image across the x-axis.
NCERT Examples
NCERT Example 1 — Identifying quadrant or axis

Question: In which quadrant or on which axis do each of the following points lie? A(1, 1), B(–2, 3), C(–3, –5), D(2, –4), E(–3, 0), F(0, –2).

Solution:

  • A(1, 1): x = +, y = + → Quadrant I
  • B(–2, 3): x = –, y = + → Quadrant II
  • C(–3, –5): x = –, y = – → Quadrant III
  • D(2, –4): x = +, y = – → Quadrant IV
  • E(–3, 0): y = 0 → point lies on the x-axis (to the left of origin)
  • F(0, –2): x = 0 → point lies on the y-axis (below the origin)
NCERT Example 2 — Writing coordinates from a description

Question: Write the coordinates of the point P shown in the following descriptions: (i) P is 4 units to the right of the y-axis and 3 units above the x-axis. (ii) P is 2 units to the left of the y-axis and 5 units below the x-axis.

Solution:

  • (i) Right of y-axis means x is positive: x = 4. Above x-axis means y is positive: y = 3. So P = (4, 3), which lies in Quadrant I.
  • (ii) Left of y-axis means x is negative: x = –2. Below x-axis means y is negative: y = –5. So P = (–2, –5), which lies in Quadrant III.
NCERT Example 3 — Plotting points and observing a shape

Question: Plot the points A(2, 5), B(–3, 5), C(–3, –2) and D(2, –2) on the Cartesian plane and state what figure ABCD forms.

Solution steps:

  1. Draw x-axis and y-axis. Choose scale: 1 unit = 1 cm.
  2. A(2, 5): 2 right, 5 up. B(–3, 5): 3 left, 5 up. C(–3, –2): 3 left, 2 down. D(2, –2): 2 right, 2 down.
  3. Join A–B–C–D–A in order.
  4. Observe: AB is horizontal (same y = 5), length = 2–(–3) = 5 units. BC is vertical (same x = –3), length = 5–(–2) = 7 units. CD is horizontal, length = 5 units. DA is vertical, length = 7 units.
  5. Opposite sides equal, all angles 90° → ABCD is a rectangle of dimensions 5 × 7.
NCERT Example 4 — Points on the same axis

Question: The points (–5, 2), (–5, –2) and (–5, 7) all share the same x-coordinate. Where do they lie, and in which quadrants?

Solution: All three have x = –5, meaning each is 5 units to the LEFT of the y-axis. They lie on the vertical line x = –5, parallel to the y-axis. Checking quadrants: (–5, 2) has y positive → Quadrant II; (–5, 7) has y positive → Quadrant II; (–5, –2) has y negative → Quadrant III. So two are in Quadrant II and one in Quadrant III, but all three are collinear (on the same vertical line).

NCERT Example 5 — Ordered pair: does order matter?

Question: Plot P(2, 3) and Q(3, 2) on the same Cartesian plane. Are they the same point?

Solution:

  • P(2, 3): move 2 right, then 3 up. Mark P in Quadrant I.
  • Q(3, 2): move 3 right, then 2 up. Mark Q in Quadrant I.
  • P is higher and slightly left of Q; they are at different positions despite using the same digits.
  • Conclusion: (2, 3) ≠ (3, 2). Order matters in an ordered pair — swapping coordinates gives a different point.
NCERT Example 6 — Reading coordinates from a graph

Question: A point P is marked on a coordinate plane. When a perpendicular is dropped from P to the x-axis, it meets at the value –3. When a perpendicular is drawn from P to the y-axis, it meets at the value 4. Write the coordinates of P and state its quadrant.

Solution: The x-coordinate (abscissa) = –3; y-coordinate (ordinate) = 4. So P = (–3, 4). Since x is negative and y is positive, P lies in Quadrant II. Distance of P from x-axis = |4| = 4 units; distance from y-axis = |–3| = 3 units.

Practice MCQs
1. The point (–3, 5) lies in which quadrant?
  1. Quadrant I
  2. Quadrant II
  3. Quadrant III
  4. Quadrant IV
Answer: (B) x is negative, y is positive → signs (–, +) → Quadrant II.
2. A point lies on the y-axis. Which of the following could be its coordinates?
  1. (3, 0)
  2. (0, –4)
  3. (–2, 5)
  4. (1, 1)
Answer: (B) A point on the y-axis has x-coordinate = 0. Only (0, –4) fits. (3, 0) is on the x-axis, not the y-axis.
3. The x-coordinate of every point on the y-axis is:
  1. 1
  2. –1
  3. 0
  4. undefined
Answer: (C) Every point on the y-axis is at zero distance from the y-axis, so its x-coordinate is always 0.
4. In which quadrant does the point (–7, –2) lie?
  1. Quadrant I
  2. Quadrant II
  3. Quadrant III
  4. Quadrant IV
Answer: (C) Both coordinates are negative → signs (–, –) → Quadrant III (bottom-left).
5. The distance of the point (4, –6) from the x-axis is:
  1. 4 units
  2. –6 units
  3. 6 units
  4. 10 units
Answer: (C) Distance from x-axis = |ordinate| = |–6| = 6 units. Distance is always non-negative.
6. The point (0, 0) lies:
  1. In Quadrant I
  2. On the x-axis only
  3. On the y-axis only
  4. At the origin, on both axes
Answer: (D) The origin lies on both axes simultaneously and belongs to no quadrant.
7. Which of these points lies in Quadrant IV?
  1. (–3, –4)
  2. (3, 4)
  3. (3, –4)
  4. (–3, 4)
Answer: (C) Quadrant IV requires x positive and y negative → (+, –) → (3, –4).
8. The points (2, 5), (2, –3) and (2, 0) all have x = 2. They lie on a line that is:
  1. Parallel to the x-axis
  2. The x-axis itself
  3. Parallel to the y-axis
  4. The y-axis itself
Answer: (C) When all points share the same x-coordinate, they lie on a vertical line — parallel to the y-axis (specifically the line x = 2).
9. If the ordinate of a point is zero, the point lies:
  1. On the y-axis
  2. At the origin
  3. On the x-axis
  4. In Quadrant III
Answer: (C) Ordinate = y-coordinate = 0 means the point is on the x-axis. (It is at the origin only if the abscissa is also 0.)
10. The abscissa of a point is –5 and it lies on the x-axis. Its coordinates are:
  1. (–5, –5)
  2. (0, –5)
  3. (–5, 0)
  4. (5, 0)
Answer: (C) "Lies on the x-axis" means y = 0. Abscissa (x) = –5. So the coordinates are (–5, 0).
Previous-Year Questions (PYQs)
PYQ 1 — (CBSE Board, 1 mark) Write the coordinates of a point that lies on the negative direction of the x-axis at a distance of 7 units from the origin.
Answer: Negative x-axis means x = –7. On the x-axis means y = 0. Coordinates: (–7, 0). This point lies on the x-axis, not in any quadrant.
PYQ 2 — (CBSE Board, 1 mark) In which quadrant does the point (–2, –3) lie?
Answer: x = –2 (negative), y = –3 (negative). Signs (–, –) match Quadrant III (bottom-left region).
PYQ 3 — (CBSE Board, 3 marks) Plot the points A(3, 2), B(–3, 2), C(–3, –2) and D(3, –2) on the Cartesian plane. Name the figure ABCD and find its area.
Solution:
  • A(3, 2): 3 right, 2 up — Quadrant I.
  • B(–3, 2): 3 left, 2 up — Quadrant II.
  • C(–3, –2): 3 left, 2 down — Quadrant III.
  • D(3, –2): 3 right, 2 down — Quadrant IV.

Length AB = distance from x = –3 to x = 3 along y = 2 = 3 – (–3) = 6 units (horizontal side).

Length BC = distance from y = 2 to y = –2 along x = –3 = 2 – (–2) = 4 units (vertical side).

ABCD has opposite sides equal (AB = DC = 6, BC = AD = 4) and all angles = 90°. It is a rectangle.

Area = length × breadth = 6 × 4 = 24 square units.

PYQ 4 — (CBSE Board, 2 marks) If the point P(k, 0) lies on the positive direction of the x-axis and point Q(0, –m) lies below the origin on the y-axis, what are the signs of k and m?
Answer:
  • P(k, 0) is on the positive x-axis, meaning it is to the right of the origin. So k is positive (k > 0).
  • Q(0, –m) is below the origin, so its y-coordinate –m must be negative, i.e. –m < 0, which means m is positive (m > 0). The coordinates of Q are $(0, -m)$ with $m > 0$, giving a negative y-value, which is below the origin.
PYQ 5 — (CBSE Board, 3 marks) The following points are vertices of a triangle: P(0, 4), Q(–3, 0) and R(3, 0). Plot these points on a Cartesian plane and find the type of triangle formed.
Solution:
  • P(0, 4): on y-axis, 4 units above origin.
  • Q(–3, 0): on x-axis, 3 units left of origin.
  • R(3, 0): on x-axis, 3 units right of origin.

QR is along the x-axis from x = –3 to x = 3: length QR = 6 units (horizontal base).

P is at (0, 4): the midpoint of QR is (0, 0) = origin. P is directly above the origin, so the perpendicular from P to base QR has length 4 units.

PQ: horizontal distance = 3, vertical = 4 → $PQ = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ units.

PR: horizontal distance = 3, vertical = 4 → $PR = \sqrt{9+16} = 5$ units.

PQ = PR = 5 units and QR = 6 units. Since two sides are equal, triangle PQR is an isosceles triangle.

9. Common Mistakes — Don't Lose Marks

  • Swapping coordinates: plotting (3, 5) where (5, 3) is required. Always read x first (horizontal) and y second (vertical).
  • Calling axis points "in a quadrant": points on the axes belong to NO quadrant. Saying B(7, 0) is in Quadrant I (where x > 0, y > 0) is wrong — y must be strictly positive, not zero.
  • Forgetting absolute values for distances: writing "distance from x-axis = –6" instead of 6. Distance is always positive; take the absolute value of the ordinate/abscissa.
  • Anti-clockwise order of quadrants: many students write the quadrant order as clockwise. The correct order I → II → III → IV is anti-clockwise (top-right → top-left → bottom-left → bottom-right).
  • Origin labelling: forgetting to label O at (0, 0) and not marking units on both axes — in board exams, these details earn presentation marks.
  • Saying x-axis is "the" axis: both axes have specific names. The horizontal one is the x-axis; the vertical one is the y-axis. Getting these mixed up in a solution loses easy marks.

10. Quick Revision Checklist

  • Coordinate Geometry invented by René Descartes → called the Cartesian system.
  • x-axis is horizontal; y-axis is vertical; they meet at the origin O = (0, 0) at 90°.
  • Every point = ordered pair (abscissa, ordinate) = (x, y); x comes first, y second.
  • Abscissa = distance from y-axis (with sign); ordinate = distance from x-axis (with sign).
  • Quadrant signs: I (+,+), II (–,+), III (–,–), IV (+,–). Numbered anti-clockwise.
  • On x-axis → y = 0. On y-axis → x = 0. On neither axis → in a quadrant.
  • Distance from x-axis = |ordinate|; distance from y-axis = |abscissa|.
  • Points with the same x-coordinate → vertical line (parallel to y-axis).
  • Points with the same y-coordinate → horizontal line (parallel to x-axis).
  • Plotting: draw axes → move along x first → then parallel to y axis → mark and label.
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