 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says find the equation of the circle passing through the points 4, 1 and 6, 5 and whose center is on the line 4x plus y equal to 16. So let us see the solution to this question. First of all we know that equation of the circle with center at the point hk and radius r is given by x minus h the whole square plus y minus k the whole square equal to r square. Now it's given to us in the question that the circle passes through the points 4, 1 and 6, 5. So putting first of all the point xy as 4, 1 in equation 1 or we call this equation a. So we put xy equal to 4, 1 in equation a and we get 4 minus h the whole square plus 1 minus k the whole square equals to r square or 16 plus h square minus 8h plus 1 plus k square minus 2k is equal to r square or h square plus k square minus 8h minus 2k 16 plus 1 is 17 plus 17 equals to r square. We call this equation 1. Now it's given to us in the question that it also passes through the point 6, 5. So again what we do is we put xy as 6, 5 in equation a and we get 6 minus h the whole square plus 5 minus k the whole square is equal to r square or 36 plus h square minus 12h plus 25 plus k square minus 10k is equal to r square or we can also write it as h square plus k square minus 12h minus 10k. Now 36 plus 25 is 61 so plus 61 is equal to r square and we call this equation 2. Now what we do is we subtract equation 2 from equation 1 and we get h square plus k square minus 8h minus 2k plus 17 minus h square plus k square minus 12h minus 10k plus 61 is equal to r square minus r square. On simplifying this we say that plus h square gets cancelled with minus h square plus k square gets cancelled with minus k square because we have a minus sign outside this bracket minus 8h plus 12h gives 4h minus 2k plus 10k gives 8k and 17 minus 61 is minus 44 equal to r square gets cancelled with minus r square that is equal to 0 and we call this equation 3. Now it's given to us that the center that is the point hk lies on 4x plus y equal to 16 therefore we can say that 4h plus k equal to 16 simply putting h and k in the place of x and y or we can say 4h plus k minus 16 equal to 0 we call this equation 4. Now again we subtract equation 4 from equation 3 and on doing this we get 4h plus 8k minus 44 minus 4h plus k minus 16 equals to 0 or 4h minus 4h is 0 plus 8k minus k is 7k minus 44 plus 16 is minus 28 equal to 0 or we can say 7k is equal to 28 dividing throughout by 7 we get k is equal to 4. Now putting k equal to 4 in equation 4 we get 4h plus 4 minus 16 equal to 0 or 4h equal to 12 or if we divide throughout by 4 we get h equal to 3. Now putting the point hk as 34 in equation 4 minus h the whole square plus 1 minus k the whole square equal to r square we get r square is equal to 4 minus 3 the whole square is 1 square plus 1 minus 4 the whole square is minus 3 square and square of minus 3 will be 9 because minus with the square sign becomes positive so 1 plus 9 that is equal to 10. Now putting the point hk as 34 and r square equal to 10 in equation of the circle given by a we get x minus 3 the whole square plus y minus 4 the whole square equals to 10 or x square plus 9 minus 6x plus y square plus 16 minus 8y minus 10 equal to 0 or x square plus y square minus 6x minus 8y plus 15 equal to 0 because 16 minus 10 gives us 6 and 6 plus minus 15 so we say that our answer to the question and required equation of the circle is x square plus y square minus 6x minus 8y plus 15 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.