Coordinate Geometry — Reflection
Introduction
Reflection is a transformation that maps a point to its mirror image across a line or point. In ICSE Class 10, you study reflection of points in the coordinate axes, the origin, and the line y = x. You also learn about invariant points — points that remain fixed under a reflection.
Reflection in the x-axis
A point P(x, y) reflected in the x-axis maps to P'(x, −y).
Mₓ: (x, y) → (x, −y)
The x-coordinate stays the same; the y-coordinate changes sign.
Reflection in the y-axis
A point P(x, y) reflected in the y-axis maps to P'(−x, y).
Mᵧ: (x, y) → (−x, y)
The y-coordinate stays the same; the x-coordinate changes sign.
Reflection in the Origin
A point P(x, y) reflected in the origin maps to P'(−x, −y).
Mₒ: (x, y) → (−x, −y)
Both coordinates change sign.
Reflection in the line y = x
A point P(x, y) reflected in the line y = x maps to P'(y, x).
Mᵧ₌ₓ: (x, y) → (y, x)
The x and y coordinates are interchanged.
Reflection in the line y = −x
A point P(x, y) reflected in the line y = −x maps to P'(−y, −x).
Mᵧ₌₋ₓ: (x, y) → (−y, −x)
Invariant Points
A point is said to be invariant under a reflection if it lies on the mirror line. Such a point maps to itself.
- Points on the x-axis (y = 0) are invariant under reflection in the x-axis.
- Points on the y-axis (x = 0) are invariant under reflection in the y-axis.
- The origin (0, 0) is invariant under reflection in the origin.
- Points on the line y = x are invariant under reflection in y = x.
Summary Table of Reflections
| Reflection in | Transformation | Invariant Points |
|---|---|---|
| x-axis | (x, y) → (x, −y) | Points with y = 0 |
| y-axis | (x, y) → (−x, y) | Points with x = 0 |
| Origin | (x, y) → (−x, −y) | Only (0, 0) |
| y = x | (x, y) → (y, x) | Points with x = y |
| y = −x | (x, y) → (−y, −x) | Points with x = −y |
Worked Examples
Example 1: Finding Reflected Points
Find the reflection of P(3, −4) in: (a) x-axis (b) y-axis (c) origin (d) y = x
Solution: (a) Mₓ(3, −4) = P₁(3, 4) (b) Mᵧ(3, −4) = P₂(−3, −4) (c) Mₒ(3, −4) = P₃(−3, 4) (d) Mᵧ₌ₓ(3, −4) = P₄(−4, 3)
Example 2: Coordinates After Two Reflections
Point P(−2, 5) is reflected in the x-axis to get P₁. P₁ is then reflected in the y-axis to get P₂. Find P₁ and P₂.
Solution: P(−2, 5) → Mₓ → P₁(−2, −5) P₁(−2, −5) → Mᵧ → P₂(2, −5)
P₁(−2, −5), P₂(2, −5)
Note: Two successive reflections in perpendicular axes (first x, then y) are equivalent to a reflection in the origin.
Example 3: Finding Invariant Points
If the point P(a, b) is invariant under reflection in the x-axis, find the relation between a and b.
Solution: If P(a, b) is invariant under Mₓ, it must lie on the mirror line (the x-axis). Points on the x-axis have y = 0. Therefore, b = 0 and a can be any real number.
Example 4: Combined Reflection
A(1, 2) is reflected in the line y = x to A'. A' is then reflected in the origin. Find the final coordinates.
Solution: A(1, 2) → Mᵧ₌ₓ → A'(2, 1) A'(2, 1) → Mₒ → A''(−2, −1)
Final coordinates: (−2, −1)
Use of Reflection in Geometry
Reflection is used to understand symmetry in geometric figures. A figure has:
- Line symmetry if it can be reflected across a line onto itself.
- Point symmetry if it can be reflected across a point onto itself.
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Confusing reflection in x-axis and y-axis | x-axis: y changes sign; y-axis: x changes sign |
| Forgetting that reflection in origin changes both signs | (x, y) → (−x, −y) |
| Confusing y = x and y = −x reflections | y = x: swap coordinates; y = −x: swap and negate both |
| Invariant point confusion | Invariant point lies ON the mirror line |
ICSE Exam Focus
Reflection carries 4–6 marks in ICSE exams. Questions include:
- Finding coordinates after reflection.
- Identifying invariant points.
- Problems with two successive reflections.
- Verifying properties of reflected figures.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Single reflection of a point | 2 |
| Successive reflections | 2 |
| Invariant points | 1 |
| Finding mirror line or axis | 1 |
Self-Test Questions
-
Find the reflection of the point (−5, 3) in (a) the x-axis, (b) the y-axis, (c) the origin.
-
The point P(4, −2) is reflected in the y-axis to P₁. P₁ is then reflected in the x-axis to P₂. Find the coordinates of P₁ and P₂.
-
Find the coordinates of the point which is invariant under reflection in the line y = x.
-
Show that the reflection of point A(a, b) in the line y = x followed by reflection in the x-axis is equivalent to a single reflection in the line y = −x.
-
If Mₓ(P) = P₁ and Mᵧ(P₁) = P₂, find the relationship between P and P₂.
In ICSE, reflection questions are often combined with coordinate geometry. Practice identifying the mirror line from a given point and its image.
