Introduction and the Cartesian Plane

The Cartesian Plane

Let's break this down. To locate a point on a flat surface, we use two number lines that cross each other at right angles. This setup is the Cartesian plane (named after René Descartes).

• The horizontal number line is the x-axis.
• The vertical number line is the y-axis.
• The point where they cross is the origin, written $O$, with coordinates $(0, 0)$.

Every point in the plane is described by an ordered pair $(x, y)$:

• $x$ (the abscissa) tells you how far to move left/right from the origin,
• $y$ (the ordinate) tells you how far to move up/down.

Key Point: The order matters — $(3, 5)$ and $(5, 3)$ are different points. Always read $(x, y)$ as 'x first, then y'.

The Four Quadrants

The two axes divide the plane into four regions called quadrants, numbered I to IV going anticlockwise from the top-right.

Quadrant	x sign	y sign	Example
I	+	+	$(3, 2)$
II	−	+	$(-3, 2)$
III	−	−	$(-3, -2)$
IV	+	−	$(3, -2)$

Key Point: Remember the sign pattern by going anticlockwise: $(+,+), (-,+), (-,-), (+,-)$.

[Board Important] A point's quadrant is decided entirely by the signs of its coordinates. A 1-mark question often asks 'In which quadrant does $(-5, 7)$ lie?' — answer: Quadrant II.

Points on the Axes

Points lying exactly on an axis are special cases:

• A point on the x-axis has the form $(x, 0)$ — its y-coordinate is 0.
• A point on the y-axis has the form $(0, y)$ — its x-coordinate is 0.
• The origin $(0, 0)$ lies on both axes.

Think of it this way: if you don't move up or down at all, you stay on the x-axis; if you don't move left or right, you stay on the y-axis.

Key Point: On the x-axis, $y = 0$; on the y-axis, $x = 0$. Points on the axes do not belong to any quadrant.

[Board Important] 'The point $(0, -4)$ lies on which axis?' — the y-axis, because $x = 0$.

Why Coordinate Geometry Matters

Coordinate geometry (also called analytic geometry) lets us study geometric shapes using algebra. Once points become number pairs, we can:

• find the distance between two points,
• find a point that divides a segment in a given ratio,
• find the area of a triangle from its vertices,
• check whether points are collinear (lie on one line).

All of these reduce to plugging coordinates into formulas — that's what makes this chapter high-scoring.

Key Point: The whole chapter is formula-driven. Master four formulas — distance, section, midpoint, and area — and you can solve almost every question.

[JEE/NEET Tip] Coordinate geometry is the foundation for straight lines, circles, and conics in Class 11 — a strong grip here pays off later.

Solved Examples

Example 1: Identify the quadrant

In which quadrant does the point $(-4, 6)$ lie?

Solution:

1. x-coordinate is negative, y-coordinate is positive.
2. Sign pattern $(-, +)$ ⇒ Quadrant II.

Final Answer: Quadrant II.

Takeaway: The quadrant depends only on the signs of the coordinates.

Example 2: Point on an axis

Where does the point $(7, 0)$ lie?

Solution:

1. The y-coordinate is 0.
2. A point with $y = 0$ lies on the x-axis.

Final Answer: On the x-axis.

Takeaway: $y = 0$ ⇒ x-axis; $x = 0$ ⇒ y-axis.

Example 3: Order matters

Are $(2, 5)$ and $(5, 2)$ the same point?

Solution:

1. $(2, 5)$ means $x = 2$, $y = 5$.
2. $(5, 2)$ means $x = 5$, $y = 2$.
3. Different coordinates ⇒ different points.

Final Answer: No, they are different points.

Takeaway: An ordered pair is read x first, then y — order cannot be swapped.

Example 4: Reading coordinates

A point is 5 units to the left of the y-axis and 3 units below the x-axis. Write its coordinates.

Solution:

1. Left of y-axis ⇒ negative x: $x = -5$.
2. Below x-axis ⇒ negative y: $y = -3$.

Final Answer: $(-5, -3)$ (Quadrant III).

Takeaway: Left/down are negative; right/up are positive.

Example 5: Origin distance idea

What are the coordinates of the origin, and what is its special property?

Solution:

1. The origin is $(0, 0)$.
2. It lies on both the x-axis and the y-axis, where the two axes meet.

Final Answer: $(0, 0)$; it lies on both axes.

Takeaway: The origin is the reference point for all coordinates.

Example 6: Sign reasoning

If a point has $x > 0$ and $y < 0$, in which quadrant is it?

Solution:

1. $x > 0$ (right), $y < 0$ (down).
2. Sign pattern $(+, -)$ ⇒ Quadrant IV.

Final Answer: Quadrant IV.

Takeaway: $(+, -)$ is the bottom-right region.

Example 7: Points with equal coordinates

Where do all points of the form $(a, a)$ with $a > 0$ lie?

Solution:

1. Both coordinates positive ⇒ Quadrant I.
2. They lie on the line $y = x$ (the diagonal through the origin).

Final Answer: In Quadrant I, on the line $y = x$.

Takeaway: Points with equal coordinates lie on the line $y = x$.

Example 8: Distance along an axis

How far apart are the points $(0, 2)$ and $(0, 7)$?

Solution:

1. Both lie on the y-axis ($x = 0$).
2. Distance along the y-axis $= 7 - 2 = 5$ units.

Final Answer: 5 units.

Takeaway: For points on the same axis, just subtract the differing coordinate.

Example 9: Mirror image in x-axis

What is the reflection of $(4, 3)$ in the x-axis?

Solution:

1. Reflecting in the x-axis keeps $x$ the same and changes the sign of $y$.
2. $(4, 3) \to (4, -3)$.

Final Answer: $(4, -3)$.

Takeaway: Reflection in the x-axis: $(x, y) \to (x, -y)$.

Example 10: Mirror image in y-axis

What is the reflection of $(4, 3)$ in the y-axis?

Solution:

1. Reflecting in the y-axis keeps $y$ the same and changes the sign of $x$.
2. $(4, 3) \to (-4, 3)$.

Final Answer: $(-4, 3)$.

Takeaway: Reflection in the y-axis: $(x, y) \to (-x, y)$.