 Hello and welcome to symptom Academy YouTube channel where we bring you every day a new question in out of problem solving for the J main and J advance aspirants. So dear students today we have got for you a question from pair of straight lines. You can say straight lines also. Let's read the question and let's find out the way to solve this question. If the straight lines joining the origin in the point of intersection of the curve. So it's a second degree curve, second degree general equation of a curve. Then there's a straight line x plus ky minus 1 equal to 0 are equally inclined to the x axis then find the value of mod k right. So guys this first draw this particular scenario. So let's take a graph here. So let's say this is our general second degree curve. Okay I'm just drawing a part of it and let's say this is a line which cuts this curve. Okay now what are we doing here we are actually joining the origin to the point of intersection of this curve by this blue straight line. So what are we doing? We are just connecting the origin to the point of intersection of the line and the general second degree equation of that curve. Okay now what we have said we have said that this pair of straight lines are equally inclined to the x axis. So right now from my figure it is not appearing to be so because I have done a rough estimate of the graph. So how do we solve these kind of questions? So dear students you would have remembered that in the pair of straight lines there is a concept called homogenization. Homogenization right what is this concept of homogenization? So this concept is very important concept where it helps you to find out the pair of straight lines connecting origin to the intersection of a second degree curve second degree general equation of a curve by a straight line right. We all know that the pair of straight line passing through the origin is a second degree homogeneous equation yes all the terms in that equation are of second degree there is no first degree there is no constant term in such an equation right. So what do we do? Using the equation of the straight line in this case which is x plus ky equal to 1 and using the equation of the curve which in our case is 5x square plus 12xy minus 6y square plus 4x minus 2y plus 3 equal to 0 we try to create a homogeneous second degree equation. So how do we do that? So all the terms which are of second degree we do not disturb them for example 5x square is of second degree so we will not be disturbing it 12xy is also of second degree so we will not be disturbing that as well. But we have 4x 4x is a degree one term so what we do we write 4x into 1 and just wait I am going to replace that one very soon and here minus 2y we will replace it with minus 2y into 1 and this plus 3 will replace with 3 into 1 square okay. So what are we doing? We are basically attempting to homogenize the second degree equation. So let's see what next comes up. So in this equation we are going to just replace this 1 with x plus ky all of you please have your attention on this. So your 1 is what? 1 is x plus ky so I am going to replace that 1 with x plus ky right and this one also x plus ky and the other one also I am going to replace that with x plus ky the whole square right okay. Now so if you see this particular equation you realize that this has become a homogeneous this has become a homogeneous second degree equation okay homogeneous second degree equation okay and a homogeneous second degree equation will always represent these green lines yes these green lines the ones which are passing through the origin yes. Now let us try to let us try to now solve the question. So in the question they have asked that this gives you a pair of straight lines which are equally inclined to the x axis guys equally inclined to the x axis means what? So this should represent a pair of straight lines which are equally inclined to the x axis equally inclined means the slope of these lines are negatives of each other for example if this is theta okay so slope of this becomes tan of theta okay. So for the other one the angle of inclination will be pi minus theta the angle of inclination will be pi minus theta so the slope of the other guy which is the blue one in this will become tan of pi minus theta which we all know is negative tan theta right. So basically students what I want to say here is that m1 and m2 would be equal in opposite in sign. Let us come back to the second degree homogeneous equation of a pair of straight lines which passes through the origin. So such a straight line actually looks like this Ax square plus 2hxy plus By square is equal to 0 right and let us say this pair is basically comprising of these two lines y equal to m1x and y equal to m2x okay and this can easily be figured out that the sum of the slopes the sum of the slopes would be minus 2h by b. So anybody who has covered the second degree homogeneous equation of a pair of straight lines would understand this fact that the sum of the slopes of the line comprising the homogeneous equation is given by minus 2h by b and the product and the product m1m2 is given by A by b but we just read the concept of m1 plus m2 so this is what we are looking out for. Now as per our given information m1 and m2 are equal in opposite in sign so can I say m1 plus m2 is a 0 and that leads to minus 2h by b also being a 0 that means my h becomes a 0 yes my dear friends h becomes a 0. So in this equation that we have over here in this equation e that we have over here as you can see right on your screen in this I will collect all the h terms what are the h terms the h terms are basically associated with the coefficients of x y yes you need to start collating the coefficients of x y and start making it 0. So let's start doing that so the coefficient of x y that I can see is basically 12 from here so let me write it down over here so 12 and we will end up getting 4k minus 2 and 6k right so it's 4k minus 2 and 6k so this should be 0 yes of course it is 2h but since h is 0 even this should be 0 which my dear friends gives you 10k plus 10 equal to 0 which clearly implies k value is negative 1 okay now what has been asked to us in the equation the question setter has asked us for mod k value yes so mod k here would be just a 1 mod k here will be just a 1 so the answer to this numerical answer type question is just a 1 I hope you enjoyed this video and learned a lot especially the concept of homogenization which is an integral part of your pair of state lines thank you so much for watching stay safe and stay healthy.