 Hello and welcome to the session my name is Mansi and I am going to help you with the following question. The question says find the equation of the circle passing through the points 2, 3 and minus 1, 1 and whose center is on the line x minus 3 y minus 11 equal to 0. So let us start with the solution to the question. First of all we know that equation of a circle with center some point h k and radius r is given by x minus h the whole square plus y minus k the whole square is equal to r square. Now it is given to us in the question that circle is passing through the point 2, 3 so since through the point x and y as 2 and 3 in this equation a xy equation a we get this h the whole square whole square is equal to r square so plus 13 equals to r square and we call this equation 1. Now it is given to us in the question that the circle also passes through the point minus 1 and 1 so putting y 1 in equation a we get h the whole square the whole square equals to r square plus 1 plus equal to r square 2 is equal to r square and we call this equation 2. Now subtracting equation 2 from equation 1 we get h minus 2 k plus 2 equals to r square minus r square which becomes minus 6 becomes minus 4 with minus h square k square gets cancelled with minus k square we have 13 minus equals to r square gets cancelled with minus r square equals to 0 we call this equation 3. Now it is given to us in the question that the center hk lies on x minus 3 y minus 11 equal to 0 so we simply put the values of x and y as h and k and we get h minus 3 k minus 11 equal to 0. Now multiplying this equation by minus 6 we have minus 6 becomes equation 4 now what we do is we subtract equation 4 equation 3 on doing this we get minus minus 6 h minus 8 equals to 0 now minus 6 6 h because minus of minus 6 h is plus 6 h first of all let us open this entire bracket we have minus 6 h minus 4 k plus 11 minus minus plus 6 h minus minus plus 18 k minus plus minus 66 equals to 0 now this implies that these two get cancelled now minus 4 k plus 18 k gives us we see that this is because minus 3 k multiplied by minus 6 gives us plus there also this is 18 k now when this plus sign comes with negative sign this becomes minus 18 k so this is k now minus 18 k with minus 4 k becomes minus 22 k 11 gives us 55 so minus 66 plus 11 gives us minus 55 equal to 0 this implies minus 22 k 5 this implies k minus 22 in the denominator here and this gives us minus 2.5 so k is equal to minus 2.5 now putting we get is minus 45 now when this goes to the right hand side this becomes just 45 and that is equal to minus 21 so we can say that h is equal to minus 21 by minus 6 because we have taken minus 6 to the denominator here minus sign gets cancelled with minus sign we have 21 by 6 is 2.5 square equals to r square we can find out the value this becomes 2 minus 3.5 the whole square equals to r square is minus 1.5 and square of minus 1.5 is 2.25 plus is 2.5 is 3 plus 2.5 that is 5.5 and square of 5.5 is 30.25 and that is equal to 0.5 5 in equation we get minus 2.5 the whole square is equal to 32.5 because equation a was x minus h the whole square plus y minus k the whole square equal to r square k is minus 2.5 and r square is 32.5 let us simplify this we get 5 plus 5 y because we see that y minus of minus 2.5 becomes y plus 2.5 the whole square and we simply put in the formula here minus 32.5 is equal to 0 plus y square minus 7x plus 5 y to 5 minus 32.5 becomes minus 14 equal to 0 so we see that our answer to this question is that the equation of circle passing through the given points and whose center is on the line x minus 3 y minus 11 equal to 0 is x square plus y square minus 7x plus 5 y minus 14 equal to 0 so this is our answer to the question I hope that you understood the question and enjoyed the session have a good day