 So let's take another one, again same thing, find vertex, focus, equation of axis, length of lattice system and the parabola this time is, let's take y plus 3 the whole square is x minus 2, so please type in first the vertex, where do you think is the vertex, so all of you are attending this session, please type where is the vertex for this parabola, alright so try to recall this standard case, so this belongs to the standard case, this comes from the standard case y square is equal to 4 A x, okay, so just see who is playing the role of what, capital Y is role is played by y plus 3, capital X role is played by x minus 2 and 4 A role is been played by 1, which clearly implies A is 1 by 4, right, A is going to be 1 by 4, now first question is vertex, vertex means x equal to 0 and y equal to 0, correct, so for vertex you would put x as 0 and y as 0, that's indicative of the 0 comma 0 vertex, so you will say similarly x minus 2 is 0, y plus 3 is 0, which means vertex would be at 2 comma minus 3, exactly, so Vidushi is correct, okay, next is focus, we know focus for such cases is at A comma 0, so I will put capital X as A and capital Y as 0, correct, so capital X is A means x minus 2 is 1 fourth and y plus 3 is 0, which means x is going to be 9 by 4 and y is going to be minus 3, so let's combine it up, let's combine it up and your focus would be at 9 by 4 comma minus 3, correct, next is equation of the directrix, equation of the directrix, so we know for y square is equal to 4 A, x equation of the directrix is x equal to minus A, so in our case it will become x is x minus 2 minus A is minus 1 by 4, in other words 4 x plus minus 7 equal to 0 would become your equation of the directrix, this become the equation of the directrix, next is equation of the axis, equation of the axis, equation of the axis we know y is equal to 0 that means y plus 3 equal to 0 is your equation of the axis plain and simple and finally the case of length of lattice rectum, length of lattice rectum is 4 A, length of lattice rectum is 4 A, lattice rectum is 4 A which is going to be 4 times 1 by 4 which is 1 unit, which is 1 unit, alright guys, so this is the critical points and critical equations for the parabola which I just now gave to you, okay, let's take one more, again find the same things, vertex, focus, equation of directrix, equation of the axis, length of lattice rectum for this parabola, sorry third one, let's take x minus 1 the whole square is let's say 36 times y plus 3, okay, again let's quickly discuss it, so this belongs to which species, so this has come from this standard form, this has come from standard form, x square is equal to 4 A y, where you can realize that the role of x is being played by x minus 1, the role of y is being played by y plus 3 and the role of 4 A is being played by 36, that means A is your 9, A is 9 in this case, okay, now vertex, when I talk about vertex, the vertex for such kind of a parabola is 0 comma 0, so we'll say x is 0, capital X is 0 and capital Y is 0, right, which means x minus 1 is 0 and y plus 3 is equal to 0, which means x is 1 and y is minus 3, which means 1 comma minus 3, 1 comma minus 3, okay, now next is focus, we know focus for such cases is at 0 comma A, so I will write x as 0 and y as A, okay, and just we'll make a role change, so x minus 1 is equal to 0 and y plus 3 is equal to 9, which means x is equal to 1 and y is equal to 6, okay, so 1 comma 6 is going to be your focus, next is equation of the directrix, equation of the directrix for such cases is y equal to minus A, right, so I will say small y plus 3 is equal to minus 9 or y plus 12 equal to 0 is the required directrix, now depart equation of the axis, we know equation of the axis for a parabola like this, x square is equal to 4 A y is your y axis itself, that means x equal to 0, so we say the equation of the desired axis is going to be x minus 1 equal to 0, and finally length of latter spectrum, length of latter spectrum is 4 A and 4 A here is going to be 36 units, 4 A is going to be 36 units, now many a times the equation that would be given to you will not be in a very straight forward manner as what I had given to you, right, for example they may give you a question like this, find vertex, focus, equation of directrix, equation of axis, length of latter spectrum for this parabola, for this parabola x square plus 8x plus 12y plus 4 equal to 0, try this out and please feel free to post your response of vertex first on the chat box, so guys in this case we have to first bring it to a generalized form, in the present scenario it is expanded okay, so we have to see first is it a quadratic in x or a quadratic in y, it is obvious that it is a quadratic in x, correct, so it would be either of the form x square is equal to 4 A y or it would be of the form x square is equal to minus 4 A y, so which of the two forms, let us find it out by completing the square, so first we will complete the square over here, for completing the square we all know that when you have a 8 over here we half it and square it and add over here, so half of 8 is 4, square of 4 is 16 and in a similar way we subtract a 16 because 16 was originally not present, so this will form x plus 4 whole square, this will form x plus 4 the whole square and rest of the terms I will take it to the other side that means I am going to take minus 12 y plus 12 on the other side okay, so minus 12 y plus 12 I will take it on the other side, now remember I have to convert it to a generalized form, generalized form is where your expression should either look like y minus k square is equal to 4 A x minus h or it should look like this or it should look like this or it should look like this, so these are the four generalized form okay, remember in all these forms your coefficient the direct coefficient of x and y is actually 1 that's why I took a minus 12 common over here, I took a minus 12 common over here, so these are your generalized forms, these are your generalized forms okay, now try to compare it with the standard case so x square this will be having as your x capital x this is going to behave as your minus 4 A and this is going to behave as your capital y okay, so now there is a role change the role changes your capital x role is being played by small x plus 4 capital y role is being played by a small y minus 1 and minus 4 A role is being played by a minus of 12 that means A is going to be 3 okay correct, so vertex we know is 0 comma 0 that is x is 0 y is 0 so you'll say x plus 4 is 0 y minus 1 equal to 0 so correct shear your answer is correct so your vertex would be at your vertex would be at minus 4 comma 1 absolutely correct next focus we know for such cases focus is at 0 comma minus A 0 comma minus A so I'll write x plus 4 as 0 and y minus 1 as minus A right minus A would be minus of 3 right so it will become minus 4 and y would become minus 2 so minus 4 comma minus 2 would be your focus is that fine next is equation of the directrix equation of the directrix so equation of the directrix for such cases is known to be y equal to A so I will say y minus 1 is equal to 3 so y minus 4 equal to 0 would be your equation of the directrix next is equation of the axis equation of the axis in this case is x equal to 0 so it simply becomes x plus 4 equal to 0 as the equation of the axis length of the latter symptom is 4a which is actually 12 units actually 12 units so I hope you guys have understood the case of generalized form of the equation of a parabola and we have done a lot of problems based on that so we'll take up a question directly find the equation of find the equation of the parabola the parabola with lattice rectum with lattice rectum joining joining 3 comma 6 and 3 comma minus 2 3 comma 6 and 3 comma minus 2 so please remember here you have been given just this information that the lattice rectum of that particular parabola is joining these two points 3 comma 6 okay 3 comma 6 and 3 comma minus 2 so let's say this pink lattice rectum so remember we could have two cases of a parabola one is like this so one as you can see in white and one could be it could be like this to find both the equations of the parabola one in red and one in white so let me give you some time you may try it out very simple case so clearly it's the case of a general form of a parabola because your axis will still be parallel to the x axis this will be parallel to the x axis so first focus on the white one all right so if I if I just ask you about the white one okay first of all the distance between these two what are the distance between these two that is four a four is going to be eight isn't it so a is going to be two correct now the midpoint of this is going to be 3 comma right so if I ask you what is going to be the vertex so you're going to say for vertex you just have to move two units along the x axis so this point is going to be 1 comma 2 remember this distance will always be a now having found the vertex and having known a we can easily write down the equation of the white parabola which is going to be y minus 2 the whole square is 8 x minus 1 so if you know the vertex and if you know the parabola is opening right words it is supposed to have the equation of the form y square is equal to 4 a x and y is now being played by y minus 2 nx is now being played by x minus 1 okay similarly for the red parabola the vertex would be at 5 comma of 2 the vertex will be at 5 comma 2 right so for such case the equation of the parabola will become y minus 2 whole square is minus 8 x minus 5 right so this solves the problem so it's a case of a shifted parabola or a generalized parabola