Students in this video, let's discuss reflection of a point about the line now before we start, let
me quickly tell you what this means.
Let me have my access over here.
So this is my Y-axis and this is my X-axis.
Now, say I have a line over here and I have a point over here.
Now, the reflection of this point about this line means I need to find a point which is of the same
length from this line as this point over here.
And it should be on the other side of the line.
That is, I need to find this point over here, which is of the same length.
If this is X, this also should be X and it should be on the other side of this line.
That is what we mean with the reflection of a point about the line.
All right.
Now let's proceed.
We will classify these type of cases into three types.
The first type is finding the reflection of a point about the line Y is equal to X or Y is equal to
minus six.
The second case is finding the reflection of a point about Y X is equal to zero or Y is equal to zero.
So these are also lines, OK?
And the third case is the general case where we find the reflection about any point.
Now, why do I classify it like this?
It's because these two cases can be solved very quickly.
Right.
You just need five seconds to find the answer.
All right.
Let's start with these two cases over here.
I have my X and Y axis.
Now, first, let me draw these four lines.
This is the line Y is equal to X. Now, why is that?
So you will see that when X is equal to one, Y also will be equal to one.
Similarly, when X is equal to Y, you also will be equal to two.
Right?
So that's why this is the nine Y is equal to X and C over here you have no Y intercept.
That's because it passes through the origin already.
Now this is the equation of why is to X Y is equal to X.
Now let's proceed and draw the equation for Y is equal to minus X.
If Y is equal to minus X, the equation will be this one.
Why is that so when X is equal to one, Y has to be minus one, right?
So X is one, why is minus one when X is equal to Y has to be minus two.
Right.
And when X is equal to minus one, Y has to be one.
So that's why this is the equation for Y is equal to minus X.
So we draw this line like this.
All right, now let's proceed.
We have seen visually how we represent Y is equal to X and Y is equal to minus X.
Now can you tell me what would be the nine X is equal to zero.
Yes, the line X is equal to zero would be the Y axis.
Right.
And similarly, the line Y is equal to zero will be the x axis.
Right.
Do you agree.
Yes, that's true.
Right.
Big.
Any point on the x axis, for example, this one let this point before zero.
You can see that Y is equal to zero.
Right.
Or take this point that this V to come over here.
Also, you can see why is equal to zero.
That's why this is the nine Y is equal to zero.
Similarly big.
Any point on the Y axis, for example, this one let it be three comma zero comma three.
Right over here you can see X is zero or take this point, let it be minus zero, comma minus four over
here.
Also you can see X zero.
Right.
Therefore this is the nine X is equal to zero and this is the nine Y is equal to zero.
Alright, now let me clear up the space.
It's always good to visually understand why these methods work so that you will be able to retain and
apply them, right.
All right.
Now we have.
Drawn the four lines over here.
OK, now let's identify a point three comma for now, we are going to find the reflection of this point
about these four lines.
All right.
We start with why is equal to it.
First, I will tell you the method that is the trick, and then we will think about it and understand
why it is true.
Now, if you want to find the reflection of three comma for about why is equal to X, all you need to
do is you need to interchange X as Y and Y as X, that is for comma three will be the reflection.
Now why is that so?
Now if you draw a perpendicular line through three comma four that's perpendicular to Y is equal to
X, it will look like this and you will see that this distance and this distance over here will be equal.
And remember, that is what we mean with reflecting a point about the line.
So the reflection of three comma for about Y is equal to Xs for comma three.
All right.
Now let's proceed and find the reflection of three comma for about Y is equal to minus X.
Now, if you want to find that all you need to do is you need to interchange why ESX and excess weight
and you need to change the science, that is the reflection of three Khama four about why is equal to
minus six is equal to minus four comma minus three.
Now the reason is the same.
If you draw a perpendicular from three comma four to one minus X.
Right, it will be like this and you will see that this point is at the same perpendicular distance
from Y is equal to minus six as this point over here.
All right.
Let's proceed.
Now, if you want to find the reflection of three comma for about X is equal to zero, right?
That's this line over here.
Let's draw a perpendicular.
That is true.
Three, four.
And it's perpendicular to X is equal to zero.
Right.
It will look like this.
Right.
That is the Y value y coordinate remains the same and you get the answer as minus three, comma four.
So you just need to find the minus of the X value and the Y value remains the same.
If you remember it using this diagram, then you will not need to buy it and you will not make a mistake.
So you keep Y the same and X becomes minus X.
And visually, this is the reason, right?
We have drawn a line perpendicular to X is equal to zero through three comma four, and it should be
at the same distance like this distance over here is equal to three.
Right.
So this distance also should be equal to three.
And because it is on the left side over here, it has to be minus three.
Right.
That's why we have minus three over here and we have minus three four, which is the reflection of three
comma for about X is equal to zero.
All right.
Now let's take the reflection of three comma for about Y is equal to zero.
Now, for that, can you tell me what you need to do?
You just need to keep X constant and take the negative value of white.
And visually, this is the reason you draw a line through three comma for that is perpendicular to Y
is equal to zero.
But that line would look like this.
Right.
And then this distance over here is equal to four.
So you need to take the same distance over here and hence you will get minus four over here because
it is in the negative y axis.
Right.
So let's quickly revise what we have learned.
We have seen what the meaning of reflecting a point about a line is and we have discussed for special
cases.
If you reflect a point about why is equal to X, you just need to interchange the value of X and Y.
If you reflect about Y is equal to minus X, you interchange and change the sine.
If you reflect about X is equal to zero, you keep the value of Y same and you change the value of X
to minus X.
Right.
And if you reflect about Y is equal to zero, you keep X same and you change Y two minus.
Right now we are left with the general case.
Now let's take up the general case in the next video.