 hello friends so welcome again to another session on gems of geometry and in this session we are going to discuss about something called coaxial circles now mind you this is not coaxial this is coaxial circles okay and what does it mean so coaxial circles are nothing but a family of circles right which are satisfied or which can be represented by this particular equation x square plus y square minus 2x plus c equals zero so if you see the center over lies on the x axis and depending upon the c you can determine the radius of the family of the circle so this you know the center has to be on one axis and radius could be anything now if we you can see there are two values here a and c and we will see different kinds of coaxial circles or family of circles then I change the value of a and c so when c equals to zero you can see the circle is tangent to the y axis or the otherwise that is the y axis the tangent to the circle now what happens if I change the value of a so right now a is minus five so if you see I am now changing the value of a so you can see all these are the coaxial circles you can also monitor the position of center o right now merges with the origin and then these are the other set of so when a is positive so these are the set of let's say coaxial circles right so this is when c equals to zero so all the circles are tangential to the y axis and the center lies on the x axis this is one case so now I've taken value of c to be three you can clearly see here there is some gap between the circle and the y axis so now y axis is not touching the circle okay so now what happens if I change the value of a so as the center moves you can see right just have a look at the center's path so now you can see there is some value of a where there is no circle right there is no circle then c is positive definitely you can see the radius is coming out to be imaginary here so hence there is no circle now the moment it crosses one particular point can you see this yeah then again circles are appearing so this is another set of coaxial circles right another set of coaxial coaxial circles right in this case the value of c is positive and a for different different values of a you get these set of circles okay so another set of coaxial circles now what happens if c is less than zero c is negative now clearly you can see the circle is intersecting d y axis at two points a and b now if I change the value of a so this is what it is going to be right so all the circles are passing through a and b and this is how all right so now the circles are again all the circles are passing through d point a and b a set of coaxial circles when c is negative okay so I hope you understood what coaxial circles are and then we will be studying some theorems related around coaxial circles and value