 Next is the concept of equation of common chord of two circles. So if you have two circles which intersect, then this is called the common chord. This line is called the common chord. Now basically you can treat this common chord as a circle of infinite radius which is passing through the meeting point of two circles. Let me call these circles as s equal to 0 and s dash equal to 0. So I mean if you want I can write it in the terms of a general second degree equation x square plus y square plus 2g1x plus 2f1y plus c1 equal to 0 and this is x square plus y square plus 2g2x plus 2f2y plus c2 equal to 0. Then the equation of this common chord will be given by s minus s dash equal to 0. Please note this down. The only way you can generate an equation of a line from the equation of two circles is when you subtract them. So when you subtract them you will get something like 2g1 minus g2x plus 2f1 minus f2y plus c1 minus c2 equal to 0. This is going to be your equation of the common chord, equation of common chord. So nothing hi-fi about this. Any question with respect to this? So you tell me when can be the common chord be of maximum length? When can the common chord be of maximum length? What can be the maximum length of the common chord? Maximum length of the common chord can only be equal to the diameter of the smaller circle. Okay? Is that fine? So let's take a simple question on this. Prove that the length of the common chord of the two circles given by c1 and c2 or you can say s1 and s2 whose equations are x minus a square y minus b square equal to c square and x minus b square plus y minus a square is equal to c square is under root of under root of 4c square minus 2 times a minus b square. Please type done if you are done with it. First tell me the equation of the common chord. What will be the equation of the common chord? Please type in in the chat box the equation of the common chord you should have got it by now. Right? Equation of the common chord as Lalita rightly pointed out x equal to y, right? So guys here is a situation where you have two circles of equal radius intersecting on the line y equal to x so this is your line y equal to x. So now the center of these two circles is very clear this is a comma b and this is b comma a. Okay? So what is the distance of the center from this line? So you say b minus a mod by root 2, correct? And this is nothing but your radius which is c. So half the length would be what? This is Pythagoras theorem so let me call it as pqr, so what is going to be qr square? qr square is going to be pr square minus pq square, okay? So qr square, pr square is c square, pq is already known to you this is your pq that is going to be b minus a square by 2, okay? So qr is going to be under root of c square minus a minus b whole square by 2. So you know length of this common chord would be length of the common chord is twice of qr. Length of common chord is twice of qr. So 2 times qr will be under root of 4c square minus 2 times a minus b square, okay? Simple, done? So next concept that we are going to discuss is the very important concept of family of circles, family of circles, okay? Now in this case we are going to discuss five cases under which we can form a family of circle situation, okay? All of you understand the meaning of the term family of circles means let's say I give you two circles and so let's say these are two circles and through the meeting point of these two circles, let's say from these the meeting point you can draw infinitely many circles, right? So one can be a circle like this, right? Another can be a circle like this, okay? So all these circles will constitute a family created by these two circles which I am calling as s equal to 0 and s dash equal to 0. So when two circles intersect that is our condition number one. When two circles intersect s equal to s equal to 0 and s dash equal to 0 then the equation of the family of circles, the equation of the general family member could be written as s plus lambda s dash equal to 0. Just remember one thing over here, if lambda becomes minus 1 that will be a chord of, that will be the common chord of these two circles. That would be the common chord of these two circles. So you can call this as a limiting case where the circle becomes a line. That means the circle having infinite radius, is that fine? So this is the first situation where when two circles having equation s equal to 0 and s dash equal to 0, this is the equation of any family member which passes through the intersection of these two circles, okay? And you would be given some additional condition to get this lambda. So lambda can be found out by some additional condition that we will be discussing while doing the problems on this. Second is a situation where you have a circle and you have a line which intersects this circle. So let's say you have a line intersecting this circle. Now even from the point of intersection of this circle and a line you can pass infinitely many circles, right? A circle can be made like this, a circle can be made like this. So all of them would now constitute a family. So let's say the equation of this is s equal to 0 and the equation of line is l equal to 0. Then we say that this is the equation of the general family member, is that fine? And the lambda would be again provided to you to get the, the additional condition will be provided to you to get the value of lambda over here, okay? Moving on to the third case. Third case is a situation where you have been given a circle and you have been given a line. But this time the line touches the circle and goes. It doesn't cut the circle, it touches the circle and go, okay? Now in this situation also the thing doesn't change, it is still the same concept. So through the point of meeting of these you can draw infinitely many circles like this, okay? Like this, okay? Like this. So all these will constitute a family of circles again whose general equation will be s plus lambda l equal to 0, okay? Fourth condition is, fourth condition is if you have been given two points, let's say I give you two points P and Q, right? X1, Y1 and X2, Y2. Now how many circles can pass through these two points will also constitute a family, right? But here you would be, here you would be surprised because you have not been given any circle so as to say and you have not been given any line or any other circle. So there is no circle given to you, there is no line given to you and still you are getting family of circles from here, right? Now you can try to visualize this condition as if you have been given a circle whose diameter is X1, Y1 and X2, Y2. So you will get a unique circle, this will be a unique circle, okay? So let's say this is given to you and you have been given a line which is exactly connecting X1, Y1 and X2, Y2. So this will be a unique line, this will be a unique line. So treat this situation as if s equal to 0 is given to you and l equal to 0 is given to you. That's as good as saying these two points are given to you and you have to form infinitely many circles passing through these two points, right? So in that case, first of all you have to find what is your s. So correct me if I am wrong, your s will be nothing but X minus X1, X minus X2, Y minus Y1, Y minus Y2, okay? This will be your s, okay? If you put it to 0 you will get the equation of the circle. Now what will be your l? l basically is nothing but a line connecting X1, Y1 and X2, Y2 which you can write in determinant form like this. So when you say l equal to 0 means you are saying this determinant is equal to 0, correct? So s equal to 0 and l equal to 0 would represent the unique circle, this would be the unique circle which I drew through X1, Y1 and X2, Y2 and this would be the unique line that you can draw through X1, Y1 and X2, Y2, okay? So by using these two, by using these two I can say the equation of any family of circles passing through these two points can be given as s plus lambda l equal to 0, right? That is X minus X1, X minus X2 plus Y minus Y1, Y minus Y1, Y minus Y2 plus lambda mod, not mod determinant X, Y1, X1, Y1, 1, X2, Y2, 1 equal to 0. Is that clear? Any questions with respect to this? So moving on to the fifth case, let's say you have been given a line, you have been given a line and you have been given a point on the line. So let's say l equal to 0 is your line, right? And you have been given, you have been given this point X1, Y1, okay? So now this is as good as saying you have a point circle. You have a point circle and you have a line meeting. Now many people ask sir, how can a line and a circle make a circle? See, line is also a circle of infinite radius. Treat line as a circle of infinite radius, okay? Because see, if you have to form a family, of course humans and human can only give rise to a human, right? So how can a circle and a line give rise to a circle? You can also treat line as a circle. Now if I ask you what would be the equation of any family member which is touching this line at this circle, okay? You would first find out the point circle equation which is nothing but X minus X1 square plus Y minus Y1 square. This will be a point circle, right? So S equal to 0 would be the equation of a point circle, okay? And equation of the line, you can always find it by using the fact that, let's say the slope of this line is given to you, equation of the line is given to you. So it would be obviously of the form Y minus Y1 minus MX minus X1, okay? So L equal to 0 would be the equation of the line. So M would be provided to you, so M will be provided to you. So again you can say this would become the equation of the family of circles in this case. Is that fine? So these are the five cases under which we can categorize the concept of family of circles.