 Now we will take some very basic questions where I will give you the parabola and I would ask you to get me various aspects or various critical points and critical equations with respect to this. So find vertex, focus, equation of directrix, equation of directrix, length of the lattice rectum or let us say equation of axis, equation of axis, length of the lattice rectum for the following parabolas, for the following parabolas. I will start with the very basic one. Let us say y square is equal to 8x, let us say y square is minus x. x square is 16y, x square is minus 8y. So very easy, just a warm up kind of a question and then we will be going to some deeper concepts. So I am giving you exactly two minutes to complete this. So while you are solving it, let us start the discussion for the first one. So the first one, the vertex is going to be at the origin. So I will keep writing the answers over here as well. Let us change the color, let me write it in blue. So vertex for the first one is 0, 0, no doubt about it. Focus would be at a, 0. Now what is a over here? 8 is 4a, so a is going to be 2. So focus is going to be at 2, 0. Equation of the directrix would be x equal to minus a. So x equal to minus 2 would be the equation of the directrix, okay. Next what is the equation of the axis? Equation of the axis would be your x axis itself which is nothing but y equal to 0, okay. Length of the lattice rectum is normally this number that you see over here, okay. So 8 units will be your length of the lattice rectum or you can say 4a is the length of the lattice rectum, okay guys. So no problem in doing the first one, it was super simple. So let me quickly rush through the other parts as well. So let's talk about the second one, y square is equal to minus x. If possible make a rough sketch in your mind, y square is equal to minus x would be like this, okay. So vertex will still be at 0, 0, right. For focus I need to get my a. Now remember here 4a is minus, 4a is equal to 1 over here. This negative sign only represents its orientation. It only represents the orientation of the parabola, okay. So 4a is still 1, so a will be 1 by 4. So your focus will be at minus a comma 0, minus a comma 0. Equation of the directrix would be x equal to a in this case which is x equal to 1 by 4. Equation of the axis will still remain the x axis which is y equal to 0 and the length of the lattice rectum is going to be 1 unit. Length of the lattice rectum is going to be 1 unit in this case. Remember whatever the coefficient of x, take mod of it that is going to be your length of the lattice rectum. So moving on to the third problem, x square is equal to 16y, x square is equal to 16y make a rough figure in your mind, it's going to be a parabola opening upwards. Vertex will still be at 0, 0, correct. Focus would be at 0 comma a, so to know your a just equate 16 to 4a, so a will be 4. So focus will be at 0 comma a. Equation of the directrix would be y is equal to minus a, y is equal to minus a. Equation of the axis is going to be your x axis, sorry y axis which is going to be x equal to 0 and length of the lattice rectum is going to be 16 as I have already discussed with you, 16 units would be the length of the lattice rectum. So now moving on to the last problem, x square is equal to minus 8y, so x square is equal to minus 8y is going to be a parabola opening downwards, however vertex will continue to be at 0 comma 0. Focus would be at 0 comma minus a, remember 4a is 8 over here. So a is 2, so 0 comma minus a would be 0 comma minus 2 which is this point. Equation of the directrix will be y is equal to a which is nothing but y is equal to 2. Equation of the axis is going to be your y axis which is x equal to 0 line and length of the lattice rectum is this particular figure over here which is 8 units. So by this we have completed the 8th question as well. Now guys, so far we were talking about such cases of a parabola whose vertex was at origin and whose axis is where exactly the x axis or the y axis, correct? Those are called standard forms. So what are standard forms? Standard forms are those cases where standard forms are those cases where the vertex is at origin and the axis of the parabola is either your x axis or your y axis, okay? Now we are going to learn something called the generalized form. We are going to learn now the generalized form. When I say generalized form, now the vertex will not be at origin, right? But your axis of the parabola would be parallel to the x axis or parallel to the y axis, okay? So let me draw a typical diagram of a generalized case of a parabola. So let's say this is our y axis, this is our x axis, okay? If I draw a parabola like this, if I draw a parabola like this where the vertex is now at some point, let's say h comma k, okay? And your axis of the parabola is now parallel to the x axis like this, okay? Then in this case such a parabola would be called as the generalized form of the parabola. So this is your axis, this is your vertex, right? Now how does the equation of this parabola change? If you would realize this, it is actually a case of shifting this parabola, right? So this dotted parabola has been shifted to this position, isn't it? Okay, can somebody tell me if you're shifting a parabola up by let's say k units and right by h units, how does the equation change? We have already done this in our bridge course. So when y square is equal to 4ax parabola is shifted h units to the right and let's say k units up, we know that the transformed equation, this gets transformed to y minus k the whole square is equal to 4ax minus h, right? You can also see this concept as shifting of origin, okay? So if you want your parabola to go up and right, then that in that case you are shifting your origin to minus h minus k, remember that, right? In that case you are shifting your origin to minus h comma minus k. So this becomes the equation of a parabola whose vertex is that h comma k, but its axis is still parallel to the x axis and it's opening towards the right, is that okay? Now in a similar way, the rest other equations will also get changed as per what was the orientation of the standard parabola which you are generalizing and where are you shifting the vertex of the parabola, okay? So I'll take just one more case so that you understand the concept well. So if I say, if I say I have drawn a parabola like this, so this distance is still a, okay? And the vertex is now at h comma k, right? And this becomes your axis of the parabola, this becomes your axis of the parabola, okay? In that case how would you write the equation of the parabola? You would say very simple, the equation of the parabola in this case would become x minus h whole square is minus 4A y minus k, okay? Now next we are going to take up some questions where I will give you a generalized version or a generalized form of a parabola and I would ask you the critical points and critical equations of that parabola. Now this is something which is beyond your school, in school exam they will never ask you a generalized version of a parabola. In the CBC curriculum we are mostly focused or we are mostly catering to the needs of standard case of a parabola. So let's take a question here, find the vertex, again same set, focus, equation of directrix, equation of axis, length of lattice rectum, length of lattice rectum, okay? For the following parabola, for the following parabola, so let me give you a simple one to start with x y minus 2 the whole square is equal to 16 x plus 1. I would just give half a minute or one minute for you to try it out and then I will solve it for you. All right, so I will just discuss how to attempt such problems. Just try to compare this with y square is equal to 4A x, right? So there is no negative sign over here, so I will not compare it with y square is equal to minus of 4A x but I will compare it with y square is equal to plus 4A x, okay? Now if you directly compare you would realize your y-roll is being played by y minus 2, okay? X-roll is being played by x plus 1 and 4A-roll is being played by 16, correct? Which means your A is going to be 4, okay? Now see the approach for solving this problem. The first problem is we have to find the vertex, okay? So just make a comparison chart. So on this side I would write y square is equal to 4A x. And on this side I would write y minus 2 whole square is equal to 4A x, 16 I have written it as 4 into 4, okay? Now we know for vertex your x and y both are 0, right? We know that coordinates of vertex is 0 comma 0, isn't it? This is your vertex which implies both x and y are 0. Now I would do a similar thing on this side as well. That means I would write x plus 1 as 0 and I would write your y minus 2 as 0, right? Which gives me x value as minus 1 and y value as 2 which means the vertex of this parabola would be at minus 1 comma 2. Is this clear guys? So this is going to be my vertex, okay? Second thing is we want to find the focus. We know focus is A comma 0, right? When you say A and 0 it means your capital X is A and your capital Y is 0, right? So this is your capital X and this is your capital Y, correct? So I am going to do the same role change over here. I would write capital X as small x plus 1 equal to A, A is 4 and I am going to write capital Y is y minus 2 equal to 0 means this is equal to 0. That means x becomes 3, y becomes 2 that means combine it and make a point out of it. So 3 comma 2 would be the position of the focus, okay? Next is we are asked the equation of the directrix, okay? So equation of directrix for such case of a parabola y square is equal to 4ax, we know that the equation is x equal to minus A, right? So here also I would say x is x plus 1 minus A is minus 4 which implies x plus 5 equal to 0 becomes the desired directrix. So this becomes the equation of the directrix. Please note it is an equation, just do not try it minus 5, this is your directrix, okay? This is equation of the axis, equation of the axis. Equation of the axis in this case is y equal to 0 because it is your x axis. So in this case you would say y minus 2 equal to 0, this becomes your desired equation of the axis of the parabola, okay? And lastly we have length of lattice rectum, length of lattice rectum is given as 4A units. So in this case it will become 4 into 4 which is 16 units, okay? So this is your length of lattice rectum. Is it clear? Shia, Yamini, Arayman, is it clear? So now guys you need to respond to the question which I am going to ask you next. So in case of any doubt, please type it on the chat box. In case you are fine with it, please type clear so that I can proceed, alright? So Shia says yes.