 
 
Hello students, today I'm here with a new topic
'Curve Tracing'
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Cayley hamilton theorem, and consistency and inconsistency
and then on partial differential, I uploaded
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and also differential equation-based topics like first-order first degree
and second-order differential equation, Fourier transform, Fourier series
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Today we will discuss curve tracing
which is a very important part
as it is a part of the differential calculus
and here in the upcoming concepts
of finding the volume of a curve, surface area or revolution of the curve or area of the curve
with the help of integration
so you should know the concept of curve tracing
as to have an idea of the curve
Without it, we can not determine any of the above things for it
It is also useful for a future topic i.e. change of order of integration
So students, today I'll teach you
How to trace a curve
There are 3-4 very important concepts in curve tracing
The first concept is 'Symmetry'
 
 
 
It is of two types- about x-axis and y-axis
or about a line
For example, you have a parabola in front of you
 
 
Students usually get confused
Which parabola is what
Now I'll tell you all about it in curve tracing
whose power is even
 
If in any curve, the power of 'y' is even
then it is symmetric about the x-axis
 
 
 
If in any curve, the power of 'x' is even
then it is symmetric about the y-axis
 
 
 
The next concept is 'Origin'
Let me tell you one more thing
for a given circle
'x' and 'y', both have an even power
When both the powers are even, then curve is symmetric about both axis
 
This circle is symmetric about both the axis- 'x-axis' as well as 'y-axis'
i.e. it divides the circle into two equals when seen from the respective axis
 
 
The next concept is 'Origin'
A curve passes through the origin
If on putting x=0 and y=0 in any curve's equation
the equation gives '0'
putting x and y as 0 in this equation gives 0 i.e. it is passing through the origin
but here equation does not give 0
hence this curve i.e. circle does not pass through the origin
So if any curve passes through the origin
then its equation should become '0' when we put x=0 and y=0
 
The next concept is tangent at origin
It means
If any curves passes through the origin
it's tangent exist there
Just like it is passing through an origin
then it has a tangent
Just like that it also has a tangent
But because this doesn't pass through an origin
it's tangent at origin doesn't exist
 
While tracing the curve we have to check  whether it passes the origin or not
 
 
 
 
Now we'll see how to calculate tangent at origin. So students if in any curve
when we place a least degree term as zero
it results in its tangent at origin.
Now this is the curve with X and Y terms
the least power is of x
because x is of 1 degree.
so the one with least degree
 
 
So this curve's tangent at origin will be y-axis
 
So the y-axis is the tangent
Which touches the curve at the origin.
Similarly in this curve
The least degree term is y
 
 
 
 
So this curve's tangent at origin will be x-axis
This will be the third point.
The next point will be intersection with coordinate axis.
 
It means that
Where does this curve cuts the
the x-axis and the y-axis.
 
 
To know where is this curve cutting x-axis
 
Then we put y=0
because at x-axis y coordinate is always zero.
So by putting one coordinate's value we can calculate other
 
 
 
 
 
 
 
Similarly if you want to where is it touching y-axis
 
so we'll keep x equals to zero
 
 
 
This is how we check where is the curve touching each axis.
 
 
 
 
 
We'll here check whether the curve is open  or closed curve
 
 
 
 
 
 
 
now comes the Reason of Existence
 
This is a circle, right,
and this is an equation, this is curve is here only.
Here you have to check the existence of the curve
That this circle is formed within this area only and there is no other existence of this curve
So here we will discuss 5-6 points
First is Symmetry,
to know if the curve is symmetric to x-axis or y-axis
Second is origin
Third is whether the curve is tangent at origin or not
 
Tangent is origin is calculated by putting least degree term as zero.
Then we check intersection with coordinate axis
to know where the curve touches the respective axis
 
 
 
 
 
 
 
 
 
The next concept is of Reason of Existence
our curve is traced with these five points  and with the help of 3-4 questions I'll explain you this concept.
In this video I'll tell you how to trace Cartesian curve
 
 
 
Here we'll take our first question.
 
the first step will be
to see the symmetry
In this question we have x and y both the variable
The power of y is  even
and the power of x is odd
and because then power of y is even
so it will be symmetric to x axis
 
 
 
 
 
 
 
you can also write it as
 
 
 
next we'll take origin
 
 
 
 
 
 
 
 
 
 
third point will be tangent at origin
 
tangent at origin means
, if a curve passes through origin
then its tangent exists , but because here the curve doesn't passes through the origin
so we'll write
 
 
 
now we check the forth point
that is  intersection
with
coordinate
axis.
we want to see, where does this curve
touches the x-axis
and where it touches y-axis
I would also explain you the concept of curve trace
this is x-axis
and this is our y-axis
till now we have studied that
this curve is symmetric to x-axis
because the power of y is even
 
 
 
now here we'll see
where it touches the x-axis
 
so here we'll put y=o
 
 
 
 
 
 
 
 
 
Now I want to know where it touches y-axis
so we'll put x=0
and you'll see the moment I write x=0
 
we can also write this curve as
 
 
 
 
when you x=0 the value of y becomes infinite it means
this curve touches y axis at infinite
And at the time of asymptote, I explained that
when any curve touches
any line at infinite then it is its asymptote
 
 
so here curve
does not
cut y-axis
next we'll talk about
asymptote
here we will see how to calculate asymtote
I told you before that normally a parallel asymtote exists here
 
so in this question if we multiply the euqation
 
 
the first thing we see is
what is the highest power of x
so the highest power of x here is 1
and its coefficient here is
 
If we solve it the result will be imagery
which there is no existence of
any asymptote
 
 
 
now we check what is the highest power of y so the highest power of y is 2
and its coefficient is x=0, then
yes we get asymptote at x=0, so we will write, equating
coefficient of
highest power
of y that is
 
 
 
 
 
 
 
 
that means y axis itself is parallel asymptote to its axis.
it means this y axis is a
parallel asymptote
about its axis.
at last we will check
reason of existence
so this is an asymptote
curve cuts at x axis, so obviously the curve that cuts
 
will go like this because it touches y axis at infinite point
and the way it is above will be the same at bottom
so the curve will be like this
now there could be many questions arising in your mind about the curve
like is it necessary that it goes here only why not the other side
then, the curve always goes in the direction of the asymptote
where curve touches asymptote is important.
finally we will see
Reason of existence
now we want to see that this curve
exists within this boundary only and not anywhere
so what you have to do is
you'll write the question in a way like
 
 
 
 
 
 
so what we will do here is, we will put the value here
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Here I discussed five points with you
first symmetry, then origin,
 
tangent to origin then intersection at cordinate axis
then asymptote and reason of existence
we'll take 1-2 more questions
Next we will see one more question here, which is
 
 
here power of x is odd and y is even
so the first concept will remain same that is
 
 
 
now if we put the value of x and y zero then
this complete equation becomes zero
that is curve passes through the origin
 
 
Next if this curves passes through the origin then its
tangent at origin exists
 
 
 
 
to calculate the value of a curve's tangent at origin then
put its least degree term as zero
 
 
 
 
 
 
 
 
 
 
 
 
so here the tangent at origin will be x-axis
I'll also show you the curve
what we know till now is,
the curve as much is one one is same is on other side
plus its tangent at origin is
x axis
 
 
now we'll see intersection with coordinate axis
 
in this we will first put y=0
in doing so the complete bracket will turn zero
and you get the value of x=0
curve passes through
curve
cut x-axis at
 
 
 
 
 
 
 
 
 
 
 
now calculate asymptote.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Now lets talk about, reason of existence
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Now this video has already got extended a lot
so I'll cover rest of the questions
in my next video
 
 
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