 अस्ब आप आप आप हैं वीड़े और वो बहुँगाँ नीवादागे ब्यदे बादुगाट्व्याँ. तोस्ब आज में वीड्यो लिए क्या आप आप आपके लिए आस्सिमटोट्स क्योगपर. आस्सिमटोटस जिस की चर्चा हम लों करते तो है, what is the definition of asymptotes so asymptotes are three types of asymptotes one is called horizontal asymptote where your asymptote is parallel to x axis one is vertical asymptote where your asymptote is parallel to y axis and third one is an incline asymptote an asymptote which is making a certain angle a certain non zero angle of course not 90 degree with the x axis so we will take some example where we will try to figure out these three cases one by one so let us begin with horizontal and vertical asymptotes alright let's find out the horizontal asymptotes for this curve so for finding the horizontal asymptotes for this curve we focus on what is the highest power of x present in this given equation so as you can see I have already pointed out x square is the highest power of x present in this expression group up your x square terms so as to know what is the coefficient of that x square term or what is the coefficient of that highest power of x present in that entire equation take this coefficient equate it to zero and there you go your job is done so once you get it to zero you end up getting y y minus one equal to zero which clearly implies y equal to zero and y equal to one so these two horizontal lines basically are the horizontal asymptotes for this equation alright now talking about vertical asymptotes for the vertical asymptotes the process is more or less the same now we identify what is the highest power of y occurring in this expression so what I am going to do I am going to find out what are the highest powers of y occurring in this expression which is clearly y square which is present at these two positions so what I am going to do is I am going to collect all these y square terms together and I have figured out that x square minus x happens to be the coefficient of y square so equate this to zero equate this to zero and that is all you need to do to get your vertical asymptotes so now as you can see this implies that x could be zero and x could be one so these two are your vertical asymptotes so this is how we end up finding our horizontal and vertical asymptotes for a given equation now friends let's look at a scenario where there is a possibility of an inclined asymptote coefficient of the highest power of y which is y cube is actually a one which is a constant this signifies that this curve doesn't have a vertical asymptote right so one thing is for sure there is no vertical asymptote there is no vertical asymptote now similarly could be figured out whether it has a horizontal asymptote or not well let's try to figure out what is the highest power of x present so as you can see the highest power of x present is x cube which has a coefficient of 2 now 2 being a constant it means there is no horizontal asymptote as well so there is no horizontal asymptote for this given curve now that only leaves us with the option that there could be inclined asymptotes now what are inclined asymptotes of course as the name itself suggests these are asymptotes which are making a certain angle with the x axis so basically a curve has a inclined asymptote when this asymptote looks like y equal to mx plus c so basically it has a non-zero slope and of course it may or may not have a y intercept so let's try to figure out if there is any inclined asymptotes over here or not and for that we need to figure out what is m and c values possible so to find m value let me take you through an algorithm now this algorithm I will not be proving in this video because it will unnecessarily stretch the video so that proof can be seen in a separate video of mine which will be coming very very soon so now I am going to take you through a simple algorithm which will help you to get the value of m and c if at all there is a inclined asymptote possible for this curve so what is the algorithm so first step is we collect all the highest degree terms together so as you can see the highest degree terms are m cube minus 2xy square minus x square y plus 2x cube in this expression we put our y as m so put y as m and x as 1 so when we do that we end up getting something like m cube minus 2m square minus m plus 2 ok we will call this as a polynomial in m of degree 3 so we will write it as 5 3 m ok now the value of m that I was talking about in this case to get the value of m what you have to do is we have to put let me write it as a second step you have to put the polynomial 5 3 m as 0 ok so when you do that you will be getting some values of m from there so let's figure it out I think this is very easily factorisable you may take m square m minus 2 minus 1 m minus 2 equal to 0 so this is actually giving you m square minus 1 times m minus 1 m minus 2 equal to 0 and this is easily factorisable this is easily factorisable ok so this throws out 3 values of m at you ok so what are the values of m what you get from here one is a 1 other is a minus 1 and the third one is a 2 now for these corresponding m we now need to figure out what is the c value ok so to get the c value there is a very interesting formula there is a very interesting algorithm again I will give this without the proof to you so c basically is found out by a very interesting formula which is c 5 3 m dash plus 5 2 m equal to 0 now you must be wondering what is 5 2 m because 5 3 m I have already discussed in step number 2 but what is 5 2 m so 5 2 m for that you need to see what are your second degree terms in this given expression so these are your second degree terms ok in this you just have to put your as m and x as 1 so this becomes your 5 2 m ok so the process is exactly the same as we used for 5 3 m so here so for 5 2 m you have to just take your second degree terms of this expression and put y as m and x as 1 so that is nothing but 3 m square minus 7 m ok now let's figure out the c values for each of these m's so we'll start with m is equal to 1 value so my c will come out by the use of this formula which I have written here I'll just simplify it a bit ok so minus 5 2 m divided by 5 3 dash m ok so minus 5 2 m means negative 3 m square minus m divided by 5 3 dash now you need to differentiate this you need to differentiate this like this and put your m value as a 1 and put your m value as a 1 so in this case the answer that you will end up getting is negative of negative 4 which is plus 4 divided by 3 minus 5 which is minus 2 so you get c value as a negative 2 ok so one of the asymptotes so one of the inclined asymptotes that you will end up getting is y is equal to mx plus c so which is nothing but y is equal to x minus 2 so this is one of your answers ok in a similar way we can also plug in value of m as minus 1 and get the second inclined asymptotes so let us do that as well let us get it over here for m is equal to minus 1 I will use the very same formula c is equal to minus 5 2 m by 5 3 dash m which is nothing but which is nothing but let me just copy this from here minus of 3 m square minus 7 m 4 divided by 3 m square minus 4 m minus 1 so putting a minus 1 will give you will give you minus 10 by by by by by by 6 right so this gives you nothing but negative 5 by 3 so y is equal to negative x negative 5 by 3 is your another inclined asymptote ok similarly I will be also using my value of m as 2 so let us see what do we get from there m as 2 I will get as negative 3 into 2 square which is 12 minus 14 upon 12 minus 8 minus 1 so that gives you around 2 divided by 3 so that gives you the third inclined asymptote as y is equal to 2x minus sorry 2x plus 2 by 3 so this is your third inclined asymptote ok now dears friends here you must be having a question and that question would arise if you think this slightly harder you see in your formula of c there is a 5 3 dash m coming over here but what if this 5 3 dash m becomes a 0 right it could very well happen that your denominator term over here may become a 0 when you put that particular value of m now when will this happen actually when will this happen when 5 3 m will have a repeated root ok so when 5 3 m has a repeated root there is a chance that your denominator becomes a 0 because when roots are repeated then that particular root will also be satisfying 5 3 m and will also be satisfying the derivative of 5 3 m that's how it happens when the roots are repeated the root not only satisfies the original equation but also satisfies the derivative of that given polynomial equation right so now we are going to take a case like that where I will tell you how does this formula of finding c gets modified if there is a repeated root occurring alright so let's look into this question now again let's start with our analysis of vertical and horizontal asymptotes does it have a vertical asymptote let's try to figure it out so for vertical asymptote we will check what is the coefficient of y cube y cube coefficient is coming out to be negative 4 so what does it mean it means no vertical asymptotes asymptotes ok so for finding horizontal asymptote we need to equate the coefficient of x cube to 0 but x cube coefficient is a constant which is 1 so that means there is no horizontal asymptote asymptotes ok now let us figure out whether it has an inclined asymptote so we will follow the same mechanism as I discussed with you in the previous problem so first I will choose the highest degree terms and highest degree terms are x cube minus 3 x square y minus 4 y cube and in that I am going to put my y as m and x as 1 so as you can see we will end up getting this guy right so this used to be our step number 1 right in step number 2 we used to equate this to 0 and see what are the values of m that will come out from here ok so in the interest of time I will actually give you the result this will give you the value of m as 1 minus half and minus half right so what do you see here the students that this has got repeated roots this has got repeated roots now let us try to figure out how do we find out the value of c when there is a repetition of roots of m happening ok so I will take you to the third step over here so if you see the very first value is 1 which is not repeated so for this we can use our normal formula and the normal formula as we all know is I will just write down the formula once again so it is c phi 3 dash m plus phi 2 m equal to 0 so that gives you c value as negative phi 2 m by phi 3 dash m ok so negative phi 2 m ok let us figure out what is phi 2 m oh there is no phi 2 m there is no phi 2 m because there is no second degree term over here so that clearly means this is going to be 0 by something correct this is going to be 0 by something so again let me just complete it for the sake of completing it so phi 3 m derivative is going to be 3 minus 12 m squared and if I put m value as a 1 over here it still ends up giving me a 0 so y is equal to x this happens to be my 1 of the inclined asymptotes ok now for repeated roots this was for m equal to 1 now for m equal to minus half which happens to be a repeated root the formula is going to change slightly how is this formula going to change all of you please see here the formula is now going to become c squared by 2 factorial phi double dash 3 m plus c by 1 factorial which you can skip writing phi 2 dash m plus phi 1 m equal to 0 this is the change in the formula that will happen instead of a linear expression in c it has now become a quadratic in c ok so from here we can easily figure out what is my value of c when m is minus half which again I am repeating is a repeated root ok so let's do the needful so c squared by 2 factorial and if I differentiate this guy I will get minus 24 m minus 24 m plus this is going to be c into 0 and my phi 1 m phi 1 m is you take your first degree terms and put x as 1 and y as m so that will become minus 1 plus m so this will become minus 1 plus m you can write 1 minus m as well ok put it to 0 and of course let's try to solve it by putting our m value as a negative half so it is going to give you c squared by 2 and this is going to be a 12 and this is going to be a minus 3 by 2 equal to 0 ok so this leads to leads to leads to c squared is equal to 1 by 4 so c is equal to plus minus half ok so from here we end up getting 2 more incliners in terms and those are y is equal to minus half x plus half and y is equal to minus half x minus half ok so these are our 1 2 and 3 incline asymptotes for this curve so i hope these videos were very informative to you please like, subscribe and share