 Now, what about the rest of the conics? What about the rest of the conics like circle, parabola, ellipse, hyperbola? What can you comment about all these things? So for that we have to understand the locus definition of all the conics. So what is the conic section? The conic section is basically locus of a point which moves in a plane in such a way in such a way that the ratio of its distance from a fixed point perpendicular distance from a fixed line is always a constant okay and this constant is known by the name of eccentricity and this fixed point it can be fixed points as well they are known by the name of focus and this fixed line is known by the name of direct x that is what we already know okay now there is this word eccentricity which comes up and this eccentricity actually helps us to differentiate between different different conics okay if e is equal to 1 it would represent a parabola okay if e is between 0 and 1 it will represent an ellipse if e is greater than 1 it will represent a hyperbola if e is equal to 0 it would represent a circle and if e is tending to infinity a very very large number then it will become a pair of straight lines. So hyperbola is a limiting case when e tends to infinity your hyperbola will start behaving as a pair of straight lines okay now how is this e connected to your second degree equation how is my e connected to the second degree equation that we had learned a little while ago so we realize that if e is equal to 1 if e is equal to 1 that means if it represents a parabola then these coefficients of these this equation would actually satisfy the condition h square is equal to AB okay now a question will arise in your mind how right we have been studying this when h square is equal to AB in this equation it will represent a parabola so how is this e equal to 1 linked to this h square equal to AB okay for that I would write the equation of a general conic okay so let's say the fixed point focus let's say it is a point which is I can say alpha comma beta or let's say h comma k let's say the fixed line or what we call as the direct tricks so let the direct tricks be lx plus my plus n equal to 0 now from the locus definition from the locus definition of a conic section this is going to represent a conic for me that is the distance from the fixed point is e times the distance of the point x comma y from this fixed line right I should not take h and k because h and k is already being used let's take alpha beta for the sake of more clarity let's say alpha beta okay here also we'll take an alpha beta okay now let us square both the sides if you square both the sides you get x minus alpha square y minus beta square is equal to e square by l square m square l square m square and lx plus my plus n whole square okay let's cross multiply l square m square times x minus alpha square y minus beta square is equal to e square and we have lx plus my plus n whole square okay now we all see that a is basically when you say h square is equal to ab h is basically the half of the coefficient of x y half the coefficient of x y a is basically the coefficient of x square b is basically the coefficient of y square right so let us focus only on these terms so if I ask you what is a your answer will be l square plus m square that is what we get from multiplying of this term with this term okay and from here I will get minus e square l square that is I get l square 1 minus e square plus m square correct if I ask you what is the coefficient b from this equation you would say coefficient of y square which is this term when it multiplies to this term that is l square m square okay and you will have a minus m square coming up from here m square e square so that is going to be l square m square 1 minus e square correct and what is your h term okay h term you would say can only be obtained from this expression correct that will be e square e square l m that would be e square l m correct so when these when you square this term you will get 2 l m x y yes or no and so the coefficient is 2 e square l m half it you will get h is that clear correct now what I will do is I will do a b minus h square okay so a b is basically product of these two minus square of this term now product of these two if you see it will become l 4 1 minus e square m 4 1 minus e square and we will get something like this l square m square 1 plus 1 minus e square whole square correct so when these two terms multiply I will get the first term when these two terms multiply I will get the second term and when you cross multiply these I will get this term okay minus e to the power 4 l square m square let's not forget this term as well so here I will get 1 minus e square l to the power 4 m to the power 4 okay and I will also get terms like l square m square 1 plus 1 plus e to the power 4 minus 2 e square okay and of course minus e to the power 4 from here correct this term this term gets cancelled so if you want you can take 1 minus e square common from both the terms so we'll end up getting something like this minus 2 l square m square okay so this becomes 2 this becomes minus 2 e square so 2 I have taken out common okay so if I take a 2 out common I will be left with this sorry it will be plus in between it will be plus in between okay now this is clearly 1 minus e square times l square plus m square whole square isn't it okay now guys listen to this explanation very very carefully in fact I will take you to the next page so we know a b a b minus h square is 1 minus e square times l square plus m square whole square so let me take you to the next page now you realize that where e is 1 when e is 1 what will happen to the right side of the expression what will happen to the right side of this expression you will say that the right hand side of the expression will become 1 minus 1 times l square plus m square whole square that is equal to 0 isn't it and that is the reason why we get h square is equal to a b okay for a parabola getting this point yes or no when e is between 0 and 1 what happens when e is between 0 and 1 what happens you would realize that 1 minus e square will be a positive term see this term would be a positive term and this term is anyway is a positive term this term is anyway is a positive term that means your a b minus h square would be positive in nature that means your h square will be less than a b and this is the condition for it to represent a ellipse this was the condition for it to represent a parabola are you getting this point so how is this e related to h square minus a b is what i am trying to explain you over here if e is greater than 1 what happens what happens this term over here will become a negative term whereas this term is going to be a positive term so negative into positive will be a negative term that means a b minus h square will be less than 0 that means if h square is greater than a b it would represent a hyperbola it would represent a hyperbola is the point clear so how these conic sections are coming it is clear to you okay and if e is equal to 0 what is happening if e is equal to 0 you would realize that your a b minus h square will be as good as l square plus m square whole square correct if you go back if you go back you would realize that when e is 1 your a itself is when e is equal to 0 when e is equal to 0 a itself becomes l square plus m square and b also becomes l square plus m square which clearly indicates that in case of a circle in case of a circle you get an additional condition that your h should be 0 this will become 0 correct yes or no your h will become 0 because your e is 0 and your a will be equal to b do remember you had learned this condition in class 11 that for a circle we have an additional condition that a should be equal to b and h would become 0 are you getting this point now for all these cases which I have discussed so far your delta should not be 0 your delta should not be 0 for all these conditions else it will become pair of straight lines is that fine so guys this is the basic introduction of a conic section