 Hello and welcome to the session. In this session, we will discuss the question which says that the state line 2x minus 3y plus 1 is equal to 0 is the tangent to the circle at 1, 1. The radius of the circle is root 13, find the equations of the circles. Now before starting the solution of this question, we should know our result. And that is for the line x plus vy plus c is equal to 0 and a point b x1 y1, the distance of the point b can be given line 2x1 plus vy1 plus c whole point square root of a square plus c. Now this result will work out as a key idea for solving out this question. We will start with the solution. Let the equation of the circle be x square plus y square plus 2y of y plus c is equal to 0. It is given in the question that the state line that is this line is the tangent to the circle at 1, 1. Since the circle is the point 1, 1. So we have now substituting x is equal to 1 and y is equal to 1 here. We get 1 plus 1 plus 2g plus 2f plus c is equal to 0. This implies plus 2f plus c is equal to minus 2. Now the equation which is given by equation number 1 and 1, 1 plus 1 is equal to 0 as equation number 2. Now the slope which is given by equation number 2 is equal to minus coefficient of x which is 2 over coefficient of y which is minus 3. So this is equal to 2 by 3. Now it is also given that the radius of the circle is root 13. Now we have at tangent to the point of constant root 13 to the perpendicular and the center which is minus 2 minus f in this result which is given in the t-idea. Now let this point to this line equal to mod of 2 minus u minus 3 into minus f plus minus 3 square which is equal to mod of minus 2g plus 3f plus 1 over upon square root of 2 is 9 and 4 is 9 plus 3f plus 1 over upon root 13 is equal to root 13 which further implies plus minus minus 2g plus 3f plus 1 over root 13 is equal to root 13 which further implies on cross multiplying minus 2g plus 3f plus 1 is equal to plus minus 30 which implies minus 2g plus 3f plus 1 is equal to plus 30 and minus 2g plus 3f plus 1 is equal to minus 30 which further implies minus 2g plus 3f is equal to 12 plus 3f is equal to minus 40. Now let us name it as equation number 3 and this as equation number 4. Now the equation of the line perpendicular f plus 1 is equal to 0 is by point slope form 1 here is equal to n into x minus x1 which is 1 the whole. Now slope of this line is 2 by 3 perpendicular to this line minus 3 by 2 that is is equal to minus 3 by 2 is equal to minus 3 by 2 into x minus 1 the whole is equal to minus 3x plus 3 which further gives equation number 5. Now this is the equation of the line through the point slope form 1 and perpendicular that means this line will pass through the center. So now the center whose coordinates are given by my equation number 5 is equal to minus 3 and y is equal to 5 and let us name it as equation number 3 2g plus 3f is equal to 12. Now multiplying this equation with 2 this equation with 3 is minus 4g is equal to 24 and minus 9g minus 6f is equal to 15. 2 is 13g these terms will be cancelled is equal to 39 which implies g is equal to minus 3. Now equal to minus 3 in equation number 3 we get minus 3 into minus 3 plus 3f is equal to 12 which implies is equal to 12 to 6 and which further gives f is equal to 2 into the point 1 1 then equal to minus 2. So let us name this equation in c is equal to minus 2 which implies minus 6 plus 4 plus c is equal to minus 2 which further gives minus 2 plus c is equal to minus 2 which implies c is equal to 0. Now this is my equation number 1 so putting 2 equal to 2 and c is equal to 0 in 1 vf plus y square plus 2 into minus 3 into x plus 2 into 2 into y to 0 which implies x square plus y square plus 4y is equal to 0. Equation number 4 is 2g plus 3f is equal to minus 40 minus 2f is equal to 5. Now multiplying this with 2 and this with 3 this implies minus 4g plus 6f is equal to minus 28 and minus 9g minus 6f is equal to 15 adding these 2 equations minus 13g and these terms will be cancelled with each other is equal to minus 13 which implies g is equal to 1. Now putting in equation number 4 which is this equation vf is equal to minus 4. Now putting g is equal to 1 plus c is equal to minus 2 vf into minus 4 plus c is equal to minus 2 which implies 2 minus equal to minus 2 which further implies c is equal to minus 2 plus 6 which is equal to 4. Now again putting g is equal to 1, f is equal to minus 4 and c is equal to 4 in 1 vf by square plus 2 into 1 into x plus 2 into minus 4 into y plus 4 is equal to 0. This implies x square plus y square minus 6x plus 4y is equal to 0 plus y square plus 2x minus 8y plus 4 is equal to 0. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed the session.