Invariant Points for Reflection in a Line is nothing a point that remains unvaried after the transformation applied to it. Any point on a graph on the line of reflection is an invariant point. Learn more about Invariant Points Under Reflection in a Line from this article. By making this page as a reference you can know about the Invariant Points on the graph on the line of reflection which helps you to score good marks in the exams.

**Also Read:**

- Reflection of a Point in a Line Parallel to the x-axis
- Reflection of a Point in a Line Parallel to the y-axis

## Invariant Points for Reflection in a Line – Definition

Invariant points means the points which lie on the line and when reflected in the line. So, only those points are invariant which lie on the y-axis.

The invariant points must have x-coordinate = 0.

Therefore, only (0, 4) is the invariant point. No points lying outside the line will be an invariant point.

### Invariant Points in Reflection in a Line Examples with Answers

**Example 1.**

Which of the following points (-2, 0), (0, -5), (3, -3) are invariant points when reflected in the x-axis?

**Solution:**

Given points are (-2,0), (0,-5) and (-3,3)

We know that

The points lying on the line are invariant points. when reflected in the line.

So, only those points are invariant which lie on the x-axis.

Hence, the invariant points must have y-coordinate = 0.

Therefore, only (-2, 0) is the invariant point.

**Example 2.**

Which of the following points (-2, 3), (0, 6), (4, -3), (-3, 6) are invariant points when reflected in the line parallel to the x-axis at a distance of 4 on the positive side of the y-axis?

**Solution:**

Given points are (-2,3), (0,6), (4,-3) and (-3,6)

We know that,

The points lying on the line are invariant points when reflected in the line.

So, only those points are invariant which are on the line parallel to the x-axis at a distance of 4 on the positive side of the y-axis.

Hence, the invariant points must have a y-coordinate = 4.

Therefore, (0, 6) and (-3, 6) are the invariant points.

**Example 3.**

Points A and B have coordinates (4,6) respectively. Find the reflection of A’ of A under reflection in the x-axis and A” of A under reflection in the y axis.

**Solution:**

Given that the coordinates are (4,6)

A’ = image of A under reflection in the x axis = (4,-6).

A” = image of A under reflection in the y axis = (-4,6).

**Example 4.**

A point P(a,b) is reflected in the x-axis to P'(6,-5). Write down the values of a and b. P” is the image of P reflected in the y axis. Write down the coordinates of P”. Find the coordinates of P”‘, when P is reflected in the line parallel to y-axis such that x = 3.

**Solution:**

A point P(a,b) is reflected in the x axis P'(6,-5)

We know that Mx(x,y) = (x,-y)

Thus, the coordinates of P are (6,5)

Hence a = 6, and b = 5.

P” = image of P reflected in the y axis = (-6,5)

P”‘ = reflection of P in the line (x = 3) = (-9,5).

**Example 5.**

Which of the following points (-1, 0), (0, -8), (2, -7) are invariant points when reflected in the x-axis?

**Solution:**

Given points are (-1,0), (0,-8) and (2,-7)

We know that

The points lying on the line are invariant points. when reflected in the line.

So, only those points are invariant which lie on the x-axis.

Hence, the invariant points must have y-coordinate = 0.

Therefore, only (-1, 0) is the invariant point.

### FAQs on Invariant Points Under Reflection in a Line

**1. Is a line of invariant points an invariant line?**

In linear transformation maps the origin to the origin.

So the origin is always an invariant point under a linear transformation. Every point on the invariant line maps to a point on the line itself.

**2. What are invariant points?**

Invariant points are points on a line or a shape which do not move when the specific transformation is applied.

**3. How do you find invariant points on a graph?**

Sketch each graph by using the key points, including invariant points, Determine the image points on the graph by square rooting the points. And locate invariant points on y = f(x) and y = g(x). When graphing the square root of a function, invariant points occur at y = 0 and y = 1.