 Hello and welcome to the session. In this session we will discuss the equation which says that, find the equation of the circle concentric with x square plus y square minus 4x minus 6 bar minus 3 is equal to 0 and which touches the y axis. Now before starting the solution of this question, we should know some results. And first is the standard form of the equation of the circle is h whole square plus y minus k whole square is equal to r square where center is hk is the radius of the circle. The general equation of the circle is x square plus y square plus 2 gx plus 2 fy plus c is equal to 0 where center is minus g minus f and thirdly for a given line ax plus dy plus c is equal to 0 and the point px1 y1 the distance from the given line that is this line is given by mod of ax1 plus dy1 plus c whole upon square root of a square plus p square. Now these results will work out as a key idea for solving out this question but with the solution. Here the equation of the circle is given to us. So given the equation of the circle plus y square minus 4x minus 6y minus 3 is equal to 0 and let us name this as 1. Now this is the general equation of the circle. Now comparing equation number 1 with the general equation of circle here 2g is equal to minus 4 is equal to minus 6 which implies g is equal to minus 2 is equal to minus. So by this result we can find the center of the circle. Therefore circle minus g minus f that means minus of minus 2 that is 2 and minus of minus 3 that is 3. So the center of the circle is the equation of the circle and which touches the yx the required circle with circle given by equation number 1 the required circle that circle the y axis whose equation squared circle with center whose coordinates are 2 3 and which touches the y axis whose equation is x is equal to 0. Now cd is the radius. Now cd is perpendicular to ab. This is the tangent to the circle and tangent is perpendicular through the point of contact. Now by using this result we can find out the distance. Here is the distance of the point c2 3. Now here the given point and the given equation of line. Now the distance cd by the formula is equal to mod of ax1 plus by1 plus c over square root of a2 plus b2. Now here let this be the point x1 by1 and this is the equation of the line ax plus by plus c is equal to 0. Now putting these values here this is equal to mod of 1 into 2 plus 0 into 3 plus 0 over square root of a2 which is 1 square plus b2. Which is 0 square. As a here is 1, b here is 0 and c here is 0, x1 is 2, y1 is equal to mod of 2 over root 1 which is equal to 2. Therefore cd which is the radius is equal to 2 units. Now using this result which is given in the key area here the center of this required circle is 2 3 the radius r of the required circle is equal to 2 units circle root square plus y minus 3 root square is equal to 2 square. Which further implies x square plus 4 minus 4x plus y square plus 9 minus 6y is equal to 4. Which further implies x square plus y square minus 4x minus 6y plus 9 plus 4 minus 4 is equal to 0. Which further implies x square plus y square minus 4x minus 6y plus 9 and here these terms will be cancelled with each other so this is equal to 0. So this is the equation of the required circle solution of the given equation. That's all for this session. Hope you all have enjoyed the session.