 Let us look into some questions based on this so that you get a better idea about how to use them first question is Find the equation of the circle Find the equation of the circle passing through Passing through 1 comma 1 points of intersection of the circles Points of intersection of the circles x square plus y square plus 13x minus 3 y equal to 0 and 2x square Plus 2 y square plus 4x minus 7 y minus 25 equal to 0 Solve this the moment you read this statement Passing through the point of intersection of circles at the same moment. What should strike you? What should strike you concept of? Family of circles concept of family of circles So again in the interest of time Let me solve this for you. I mean, I know it's easy So you'll say let the circle let the desired circle be Let the required circle be x square plus y square Plus 13x minus 3 y plus lambda times First convert it to a general form Even if you don't convert it into a general form, it will not you know create problem for you So you can also stick with this Okay. Now this circle passes through 1 comma 1 So this information is provided for us to get the value of lambda It's a generic case that happens with all the family of you know curve situation So put 1 comma 1 in place of x and y. So it'll be 1 plus 1 plus 13 minus 3 Plus lambda 2 plus 2 plus 4 minus 7 minus 25 equal to 0 that gives you 12 plus lambda and we'll have Minus 24 equal to 0 So that means lambda is equal to half lambda is equal to half So put your lambda half over here So put this put this thing into this place Okay, so you get the desired equation as x square plus y square plus 13x minus 3 y When you half it all it becomes x square y square 2x minus 7 by 2 y minus 25 by 2 equal to 0 a Good idea would have been to multiply throughout with 2 a Good idea would have been to multiply throughout with 2 because dealing with fraction is slightly cumbersome So multiply throughout with 2 so x square plus y square plus 13x minus 3 y plus half 2x square 2 y square plus 4x minus 7 y minus 25 equal to 0 multiply with 2 so this 2 will go off from here and I will have to coming up over here Now open this it become 4x square 4 y square or 13 x sorry 26 plus 4 30 x minus 6 y Minus 7 y is minus 13 y and you'll have minus 25 equal to 0. So this is your desired answer This is your desired Answer is that clear guys these types CLR on your on your chat box if it is clear so that we can take up the next problem Next question is going to be this question find the equation of the circle find the equation of the circle through points of intersection through points of intersection of this circle and this line which touches which touches x plus 2 y equal to 0 The question is clear guys So we have to find that circle which passes through the intersection of this line in this circle and Which touches which touches this circle please type in the answer in the chat box if you're done Anybody done so we'll start with the fact that let the desired circle be this So when we collect the terms light terms, we'll get x square plus y square plus lambda minus 2x plus 2 lambda minus 4 y Plus 4 minus 4 lambda which is 4 1 minus lambda equal to 0 Now this line x plus 2 y equal to 0 touches this circle so it touches this circle That means if you substitute x is 2 y it will only should have only one solution, right? So if you replace your x with minus 2 y You will get a quadratic equation which should have only one solution Because it is touching it Because the line is touching it. Okay, so this quadratic that you get let me simplify it first So you'll get 5 y square and Minus 2 lambda plus 4. Oh, this will get cancelled awesome So this should have only only one solution if it is only only one solution What does it mean guys? It means that this term should have been zero Which implies lambda should have been one correct So if lambda has to be one then what would be your desired equation? Your desired equation would now become x square plus y square minus 2x minus 4 y plus 4 Plus 1 times x plus 2 y minus 4 equal to 0 That gives you the equation as x square plus y square minus x minus 2 y equal to 0 So this is your answer This is your answer Is that fine? Now moving on to the next concept, which is the concept of angle of intersection of two circles Angle of intersection of two circles angle of intersection of two circles Okay, so let's say we have two circles like this. Okay, and they meet at this point They meet at this point Okay, so let us draw tangents at these points Now I have to find out the angle of intersection between these two circles Let's say I call it as theta now guys Let me connect Let me connect the point of intersection with the centers Let's say this and let's say, okay Now we all know that since we are connecting the center, let's say C1 and C2 And this is a tangent to this circle this will be 90 degree and This also will be 90 degree right and Let's say this angle here is alpha This angle here is alpha Okay So from the given diagram From the given diagram, I can conclude that theta plus 90 plus alpha plus 90 is going to be 360 degrees because they complete the full circle right Which means theta plus alpha is going to be 180 degrees Right Which means alpha is equal to 180 degree minus theta Okay, if you focus on this triangle if you focus on this triangle a C1 C2 Please focus on this triangle a C1 C2 a C1 C2 This is R1 and this is R2 and this is the distance between them This is the distance between the centers of the two circle, right? correct, so can I apply Can I apply cosine rule at a? that is cos alpha Which is nothing but cos 180 minus theta, right? So cos of alpha, let me write it like this cos alpha, which is cos of 180 minus theta Can I say it is going to be R1 square plus R2 square minus d square by 2 R1 R2 Right, which means Cos of pi minus theta we all know is minus cos theta So minus cos theta will be R1 square plus R2 square minus d square by 2 R1 R2 That implies cos of theta is going to be cos of theta is going to be d square minus R1 square plus R2 square by 2 R1 R2 So this is the angle of intersection. So this gives you the angle of intersection Angle of intersection between the two given circles any question regarding this Please type No, if there is no question now the condition for orthogonality of two circles The condition for orthogonality, you know that theta is going to be 90 degree if it is orthogonal That means cos of 90 is going to be 0 Which is nothing which implies that d square should be R1 square plus R2 square. So the distance between the two circles should be Equal to distance square should be equal to R1 square plus R2 square for the two circles to be Orthogonal to each other now. How do we see this in terms of their equations? So let us say Let us say the two circles given to us are Having the equation x square y square plus 2 g1 x plus 2 f1 y plus c1 equal to 0 and The other one that is this one and this one has the equation x square plus y square plus 2 g2 x plus 2 f2 y plus c2 equal to 0 Okay, okay Now what is d? d is basically the distance between the centers Correct and the centers is minus g1f Minus g1 minus f1 and minus g2 minus f2 Correct, so can I say d square will be G1 minus g2 square Plus f1 minus f2 square That would be the distance between their centers right and What is R1 square R1 square is g1 square plus f1 square minus c1 and R2 square is g2 square plus f2 square minus c2 Right simple radius of the general form of the equation of a circle right Any question so far guys Now just put it into this So when you expand it you get g1 square plus g2 square minus 2 g1 g2 minus 2 g1 g2 Plus f1 square plus f2 square minus 2 f1 f2 is equal to g1 square f1 square minus Sorry, this is c2 Minus c1 plus g2 square f2 square minus c2 So g1 square g1 square gone G2 square g2 square gone f1 square f1 square gone f2 square f2 square what so you'll have 2 g1 g2 plus 2 f1 f2 is equal to c1 plus c2 that is the condition for orthogonality of two circles Please remember this very very important is that fine guys