 Hello all welcome to Centrum Academy YouTube live session on parabola. So Those who have joined in the session request you to type your name in the chat box so that I know who all are attending the session So this session is going to be the very very basic of the parabola We are going to just introduce parabola here in this session and of course We'll be solving a lot of problems on the basic concepts of parabola So those who have joined in the session I would request you again Please type in your name in the chat box. All right, so we'll begin the session. I'm assuming many of you are traveling right now So I'll start with the very basic definition of a parabola. So how do we define a parabola as a locus So we'll start with the parabola definition So parabola is defined as the locus of a point locus of a point Which moves in a plane? Which moves in a plane? such that is Distance from a fixed point is Distance from a fixed point is always equal is always equal to its distance From a fixed line from a fixed line Okay, this fixed point is actually called as the focus and This fixed line is actually called the directrix Okay, now by the use of this definition, I would be actually deriving the Equation of a parabola, which we call as the standard case of a parabola But before that I would just appraise you of the basic Termologies that we used when we are referring to a parabola Okay, so let's say This is my fixed line The line which you can see in the red and this is the fixed point The point which you can see in your white Okay Let's take this point to be a point a comma zero for the purpose of simplicity and Let's take this directrix to be the line x equal to minus of a right and And this point here, I would be taking as the origin that is the meeting of the x and the y axis Now how is basically a parabola defined it's the locus of all such point h comma k Which moves in such a way that it's distance from this fixed line That is this distance. Let me call it as p m is Same as the distance from The fixed point. Let's call p s Okay, so as per the locus definition P m should be equal to p s Okay, now by using the concept of locus we can derive the equation of this case of a parabola Which we normally call as a standard parabola So we take the distance p s as h minus a the whole square k minus zero the whole square under root and The distance p m is nothing going is nothing but h Plus a mod When we square both the sides we get h minus a square plus k square is equal to h square plus a square plus 2 hk on expanding the terms we Get this expression Where h square and h square and a square and a square can be cancelled from both the sides giving us the equation as Giving us the equation as k square Sorry, not hk. It will be h a yeah So giving the equation as k square is equal to 4 a h Which we can generalize as Which we can generalize as Y square is equal to 4 a x Y square is equal to 4 a x Okay so this is the case of a standard parabola you will realize that be referred to Four cases of a parabola as standard parabola and one of them is y square is equal to 4 a x The other three I'll be telling you in some time But before that few terms that we need to be aware of for example This particular line which divides the parabola into two symmetrical halves is called the axis of the parabola Okay, of course s is called the focus This red line is called the directrix This point 0 comma 0 in this case. It is called the vertex Align connecting any two points on the parabola that is called the Let's say I connect any these two points. This will be called as the cord So this I'll be calling as the cord of the parabola. Okay a line which is perpendicular to the Axis of a parabola and connecting two points. This line is called the double ordinate this point is called the double ordinate and a line which is passing through the focus of the parabola and connecting any two points as you can see I've done it in pink That will be called the focal cord Okay, so you can say focal cord is a cord which passes through the focus Cord which passes through the focus passes through the focus Okay, and finally and finally The cord which is perpendicular to the axis and passes through the focus that is called the lattice rectum. This is the most important Lattice rectum Shortest line in Greek it means shortest line Why it is called the shortest line is because later you will learn Lattice rectum is basically the shortest focal cord Lattice rectum is the shortest Focal cord Right, so you can say lattice rectum is nothing but a Double ordinate which is actually also the focal cord or a focal cord, which is also the double ordinate Okay, so these are the terms that normally we use while Referring to while we are referring to a parabola Now I will introduce you to other forms of the parabola as well So let's talk about other forms of the parabola. So a parabola which opens to the left side Okay, so such a case of a parabola which opens to the left side Right It has its focus at minus a comma zero and Its directrix is x equal to a So its directrix is given as x equal to a Okay, and of course the axis is the x-axis itself and the vertex is at origin Okay, so such a parabola has the equation such a parabola has the equation y square is equal to minus 4 a x So the parabola has the equation y square is equal to minus 4x remember in all cases of a parabola a is always positive so always remember this We always keep our a as positive no matter whatever is the parabola we are dealing with Normally a is something which we used to donate Which we used to denote the distance of vertex from the focus right or It's the distance between the vertex and the focus hence we always keep it as a positive distance So this is the second case of a parabola This is also a case of a standard parabola Coming to the third case Third case is the parabola where the parabola will open upwards okay, in such case in such case the focus would be at 0 comma a This would be your focus 0 comma a and Directrix will be the line The directrix will be the line y is equal to minus a okay And the vertex is 0 comma 0 again The vertex is here 0 comma 0 again, but this time the axis of the parabola is the y axis So here this was the axis of the parabola in this case the axis of the parabola will be your y axis Okay, such a parabola equation is given as x square is equal to 4 a y Now you realize that this parabola equation is obtained from the first parabola equation by replacing x with y That means you're trying to reflect The previous parabola basically this is obtained by This is obtained by reflecting Reflecting y square is equal to 4 a x parabola about The line y equal to x about the line y equal to x Okay, so whenever you reflect something about the line y equal to x I'll just tell you a thumb rule what we do is we interchange the position of x and y so moving on to the fourth case in This case the parabola will be the one which opens downwards The ones which the one which opens downwards Like this Okay, so here your vertex is still at origin focus now will become 0 comma minus a and The equation of the directrix The equation of the directrix will become y is equal to a Note that again y axis is the axis of the parabola Okay, and in such cases the equation of the parabola would be given as x square is equal to minus 4 a y x square is equal to minus 4 a y So I would like you all to remember these four cases very clearly in your mind because I'll be doing the general equation of a parabola These standard cases so please remember all the four standard cases of a parabola Okay Meanwhile, we'll just start solving some problems My first problem would be based on finding the equation of any parabola in general right, so follow the basic definition which is The locus of a point whose distance from a fixed point is same as The distance from a fixed line, so I'll be starting with problem based on The locus definition of a parabola, so I'll start with the very first question Find the equation of the parabola Find the equation of the parabola whose focus is at whose focus is at minus 1 comma minus 2 Directrix is x minus 2 y plus 3 equal to 0 So please try this out. I'm giving you one minute time to try it out. So let's discuss this question So let me draw a case of a parabola whose focus is at minus 1 comma minus 2 So minus 1 comma minus 2 will be a line point over here Okay, and This is going to be a line with the positive slope slope is going to be half Are cutting the x-axis at minus 3 so somewhat like this would be the directrix So somewhat like this Okay, and we have to draw a parabola like this. So your parabola would be of This nature as you can see this is the axis of the parabola as you can see the dotted white line Okay, axis is always perpendicular to the directrix. So a couple of things that you should know Axis is always perpendicular to the directrix number one Axis is always perpendicular to the directrix Okay, number two The vertex is always the midpoint between the focus and The point where the axis meets the directrix So V is the midpoint of V is the midpoint of S and N The vertex is always the midpoint of the focus and the point where the axis meets the directrix. Okay However, these things may not be useful in solving this particular problem, but this should be known You should be aware of this Okay Now here I will use the basic definition and say let the point Whose locus I am finding is H comma K Okay, so this distance and This distance must be same. So SP should be equal to PM SP should be equal to PM. So what is SP? SP would be 8 plus 1 the whole To the whole square under root is Is equal to PM PM is the distance of a point from a line and we have already learned this given as And divide by the square of the coefficient of a line Okay, so it becomes H minus Plus three Yeah, more by under root of One square plus two square Okay So if I square both the sides, I get H plus 1 the whole square K plus 2 the whole square is Equal to H minus 2 K plus 3 the whole square By 5 so let us simplify this so it becomes 5 times H plus 1 whole square square Square a square plus 9 minus 4 HK Plus 6 H minus 12 K on for the vacation. This will give me five times H square 10 H plus 5 Plus 5 K square Plus 20 K Plus 20 equal to H square plus 4 K square plus 9 minus 4 HK Plus 6 H minus 12 K. Let's simplify this so I'll get a 4 H square right by taking care of these two terms, right and 4 K square and 5 K square will leave me with a K square term Okay, then I will have a 4 HK term coming from here Then I'll have a 4 H term coming from these two terms Then I'll have 32 K term coming from these two terms and then finally I'll have plus 16 Now this is the relationship between H and K and now we need to generalize this now we need to Generalize this to get the equation of our parabola 4 Plus 4 H to Y plus 16 equal to 0 right Mission of our minus comma minus 1 comma minus 2 and directrix is X minus 2 Y plus C So as you can see guys, it can be as complicated as this Okay, so we had seen the equation of our four standard cases That is y square is equal to 4a x y is equal y square is equal to minus 4a x X square is equal to 4a y x square is equal to minus 4a y and Here in front of you you have Another version of a equation of a parabola which is very very complicated one, right? So remember all the conic Just remember all the conic Equation is basically of this nature a x square plus 2 H x y Plus b y square Plus 2 g x plus 2 f y plus c equal to 0 Okay, and in particular When you're dealing with a parabola, I have already told you in the very first class in case of a parabola H square should be equal to a b. Is it happening in this case? Please verify H H is going to be this is 2 H. So H is going to be 2 So 2 square is equal to a b. This is your a and this one is your b So 4 into 1 so that is happening. So that's the further verification that we can do For this case, okay, so this is the case of a parabola Because it is satisfying h square equal to a b condition on the general equation of a parabola, okay In a similar line