 Alright so guys welcome back hope you have all joined the new link okay so the next question that we are going to solve here is this one if the angle between the straight lines joining foci and the ends of the minor axis and the ends of the minor axis and ends of the minor axis of the ellipse x square by a square plus y square by b square equal to 1 is 90 degrees is 90 degrees then find e then find the eccentricity of the ellipse so basically this is the situation where you join the asset when you join the foci to the ends of the minor axis let's say like this this is going to be 90 degree then find e then find e absolutely correct Gaurav very good so answer as Gaurav rightly pointed out is 1 by root 2 so guys it's very simple problem if you look at it right this is going to be 45 degree each 45 degree 45 degree each okay so this will also be 45 so as a result what happens this triangle is an isosceles triangle so this and this side will become equal correct so b becomes equal to a right so e becomes b by a e becomes b by a and we already know e square is 1 minus b square by a square so e square is 1 minus e square so 2 e square is 1 that means e square is equal to half so e is 1 by root 2 e is 1 by root 2 okay easy question next is find the find the equation of the ellipse find the equation of the ellipse with center at the origin whose minor axis whose minor axis is equal to the distance between the foci the distance between the foci and lattice rectum length is length of lattice rectum is 10 units now I can mention any one of the two equations that you feel like please type in in the chat box once you're done so you may assume your a to be greater than b in this case so you can assume a to be greater than b okay so what are the first thing given to us the first thing given to us is to be is equal to 2 ae I just the length of the minor axis is equal to the distance between the foci which means b is equal to a and second thing given to us is to be square by a is 10 correct that is b square is 5 a as you can cancel over here okay and b square is also known to be a square 1 minus e square correct now first of all let us substitute b as ae over here so it becomes a square e square as a square 1 minus e square which means 2e square is 1 that means e is 1 by root 2 okay no doubt about that and when we substitute it over here you get your b as a by root 2 that means b square is a square by b square is equal to a square by 2 okay now b square is also 5 a so 5 a is equal to a square by 2 so it implies a is equal to 10 absolutely correct most of you have given the right answer a is equal to 10 and if a is equal to 10 b is equal to 10 by root 2 so you can write the equation of the ellipse as x square by a square plus y square by b square equal to 1 which means x square by 100 plus y square by 100 into 2 is equal to 1 so x square plus 2 y square is equal to 100 so this is the desired equation absolutely correct so most of you have answered it correctly next question is a bit of application-based question so there's an arc in the form of a semi ellipse in the form of a semi ellipse it is 8 meters wide and 2 meters high at the center it is 8 meters wide and 2 meters high at the center find the height of the arc find the height of the arc at a point 1.5 meters from one end find the height of the arc at a point 1.5 meters from one end so it's a scenario like this this entire span is 8 meters wide and this is 2 meters wide so they are asking you the height of the arc they are asking you the height of the arc at a distance of 1.5 meters so they are asking you h once you're done please type in your response in the chat box 0.8 can you give in give me in terms of fractions okay 1.56 so shear is correct 1.56 yes Gaurav is also correct okay so first we'll have to give it a form of a equation of ellipse so assume that your 2 meters is your b and a is 4 okay so x square by a square plus y square by b square is equal to 1 is the equation of the ellipse okay assuming origin to be at this point so this point will be at a distance of 2.5 okay so when x is 2.5 what is y is what has been asked or you can say when x is 2.5 y is h so let's find it out so 2.5 square by 16 plus h square by 4 is equal to 1 2.5 is like 25 by 2.5 square is 25 by 4 so 25 by 64 is h square by 4 equal to 1 so you can simplify it h square by 4 is equal to 39 by 64 cancel of a factor of 4 so you get h as under root 39 by 4 under root 39 by 4 meters that's yes approximately 1.56 meters so guys since not much of a time is left I would not start a new thing I'll just keep on doing some more problems okay and whatever we have done today is basically very very important for your school level stuff as well so let's take another question find the equation of the ellipse find the equation of the ellipse whose foci is at or whose foci are 2 comma 3 and minus 2 comma 3 whose foci is or whose foci are at 2 comma 3 and minus 2 comma 3 no no other information is given okay I think you can also take whose semi minor axis is semi minor axis is a root 5 you can include this information as well as there will be so many cases so this information is also there so I have made the required correction so also include this as the part semi minor axis is root 5 now can you try once you're done you can either send me the snapshot on my WhatsApp chat or you can type it in the chat box here itself okay so let's discuss this now so guys a perfect recipe to solve this question will be by using the second definition of an ellipse which we had learned a little while ago with respect to the focal distances of any point on the ellipse so the sum of the focal distances of any point on the ellipse will always be equal to the length of the major axis right now how do I get to a because what is given to us is your b b is root 5 right now first of all the distance between the two for size 2 ae right which is clearly 4 correct so ae is equal to 2 and you know that b square is a square minus a square e square so b square is a square minus 4 and this is equal to 5 so a square is equal to 9 so your a is actually 3 okay now I would use the distance I'll use the basic locus of an ellipse that is distance from 2 comma 3 and minus 2 comma 3 is always equal to 2 a which is 6 so using the definition of an ellipse that s1 p plus s2 p let's say this is p this is s2 this is s1 s2p is equal to 2 a which is 6 I can write under root of h minus 2 square y minus 3 square plus under root of h plus 2 square y minus 3 square is equal to 6 now how to simplify this very simple you can take this term to the other side and square both the sides and square both the sides that will give you h minus 2 square y minus 3 square is equal to 36 plus h plus 2 square y minus 3 square minus 12 under root of h plus 2 square y minus 3 square so these two terms will get cancelled off and if I'm not wrong that will leave you with h square minus 4 h plus 4 and here you have 36 h square minus plus 4 h plus 4 minus 12 under root of h plus 2 whole square y minus 3 whole square okay so h square h square gets cancelled 4 4 squares also gets cancelled so you will have minus 8 h is equal to 36 minus 12 under root of h plus 2 square y minus 3 square so divide throughout with 4 you will get 2 h minus sorry 2 h plus 9 whole square is equal to 9 times h plus 2 whole square y minus 3 whole square so what I did is I brought this 36 to the left side and I divided throughout with minus 4 and I squared both the sides okay so when we do that we get a 4 h square plus 36 h plus 81 is equal to sorry this is why I'm writing y over here I should be writing okay let me write x everywhere since I've already written y in half the places so this is k this is k so 9 times h square plus k square plus 4 h plus 4 plus 9 minus 6k so when you simplify this you get 5 h square 9 k square minus 54 k I think h and h will get cancelled both will have 36 h and constant will be again 36 will be left so we can generalize this we can generalize this as 5 x square 9 y square minus 54 y plus 36 equal to 0 now this is one way to do it another way is you you would realize when you draw the figure that your your fosai is at 2 comma 3 and minus 2 comma 3 so your fosai will be parallel to the y axis so this is your scenario so this is your ellipse so this line will be parallel to the x axis okay so you can always find the midpoint first the midpoint is going to be minus 2 comma 3 and 2 comma 3 the midpoint is going to be 0 comma 3 correct and you know 2a e so you know your a and b first of all you know b is 3 and a is a root 5 sorry a is a is 3 and b is root 5 a is 3 and b is root 5 so you can also use the formula x minus 0 whole square by a square plus y minus 3 whole square by b square equal to 1 that's x square by 9 y minus 3 whole square by 5 equal to 1 that actually gives you 5 x square plus 9 y minus 3 square is equal to 45 that's 5 x square 9 y square minus 54 y plus 36 equal to 0 now this is a much easier way to solve the problem as compared to the previous one so it's an easier method okay but the reason why I showed you the other method is you can also use the other method in case you require it is that clear guys any question with respect to this so the next class that we are going to have on ellipse would be a one-hour class not more than that where I will talk about the following things how a line intersects or condition for the intersection of a line condition of intersection of a line y equal to mx plus c with the standard case of a ellipse okay and while we are doing this we'll also find out the condition of tangency we'll also find out the condition of tangency condition of tangency a second thing that we'll talk about is the equation of equation of tangents and normals okay both in Cartesian both in slope form point form and parametric form so this would be the agenda for the next class and after that I am going to start with hyperbola so request those who have not seen the hyper parabola and ellipse video today to please see it before you come for the next class because these things will not be repeated again of course doubts can be entertained okay so guys over and out from my side thank you so much for coming life bye bye have a good night