+216 Let x and y be real numbers. If x and y satisfy
x^2 + y^2 = 4x - 8y + 17x - 5y + 25,
then find the largest possible value of x. Give your answer in exact form using radicals, simplified as far as possible.
+1952 First of all, let's move almost all the variables to one side of the equation and combine all the like terms.
We get x2−21x+y2+13y=25.
Now, we complete the square for both x and y on the right side of the equation.
x2−21x+441/4+y2+13y+169/4=25+441/4+169/4
(x−21/2)2+(y+13/2)2=355/2
Wait! This looks like the equation for a circle!
Indeed, we have a circle at center (21/2,−13/2) with radius √355/2.
The value of the largest possible x is just the radius added onto the x value of the center.
x=(21/2)+√355/2
Thanks! :)
+1952 First of all, let's move almost all the variables to one side of the equation and combine all the like terms.
We get x2−21x+y2+13y=25.
Now, we complete the square for both x and y on the right side of the equation.
x2−21x+441/4+y2+13y+169/4=25+441/4+169/4
(x−21/2)2+(y+13/2)2=355/2
Wait! This looks like the equation for a circle!
Indeed, we have a circle at center (21/2,−13/2) with radius √355/2.
The value of the largest possible x is just the radius added onto the x value of the center.
x=(21/2)+√355/2
Thanks! :)
