00:01
In this question, equation of sphere x square plus y square plus z square equals to 9 is given and the point should be farthest from the given point 13 comma minus 12 comma 1.
00:18
Now, here let the function g x comma y comma z that is x square plus y square plus z square minus 1.
00:34
Now, using the lagrange multiplier, so using the lagrangian function, using the lagrangian function, this will be written as capital f equals to small f plus lambda times of g.
00:54
So, capital f will be x minus 13 x minus 13 whole square plus y minus it is minus 12.
01:05
So, y plus 12 whole square and plus z minus 1 whole square plus lambda times of x square plus y square plus z square minus 9.
01:23
Now, partial derivative of this function with respect to x.
01:28
So, this will be twice of x minus 13 and plus 2 lambda x and this will be equals to 0.
01:41
So, from here 2 x minus 26 plus 2 lambda x is equals to 0.
01:50
Adding both side by 2, so x times of 1 plus lambda which is equals to 13.
01:58
So, from here lambda will be equals to 1 plus lambda that is 13 by x.
02:06
So, lambda will be equals to 13 by x minus 1.
02:12
So, this is 1 of the equation.
02:15
Then partial derivative of the given function with respect to y and that should also be equals to 0.
02:23
So, from here partial derivative of this function with respect to y will be 2 y plus 12 that will be twice of y plus 12 and plus lambda times of 2 y.
02:38
So, from here plus lambda times of 2 y and this should be equals to 2.
02:47
So, y plus 12 plus lambda y this is equals to 0.
02:54
So, from here taking y common so this will be 1 plus lambda and this is equals to minus 12.
03:04
So, 1 plus lambda will be minus 12 upon y.
03:09
So, from here lambda equals to minus 1 minus 12 by y.
03:16
Now, the next partial derivative of the given function with respect to z and this is also equals to 0.
03:26
So, here partial derivative of this function with respect to z will be twice of z minus 1 that will be twice of z twice of z minus 1 plus 2 lambda z and this is also equals to 0.
03:44
So, z minus 1 plus lambda z is 0.
03:48
So, taking z common 1 plus lambda equals to 1...