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# Coordinates

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Let \$x\$ and \$y\$ be real numbers such that \$x^2 + y^2 = 4x + 12y.\$ Find the largest possible value of \$x + y.\$ Give your answer in exact form using radicals, simplified as far as possible.

Jan 14, 2024

#1
+36919
+1

Find max of   x+y    start with standard circle form

x^2 -4x    + y^2 - 12y   = 0

(x-2)^2    + (y-6)^2   =  4 + 36

(x-2)^2   + (y-6)^2 =  40

Now find the 45 degree line that goes through the center   (2,6)

slope    = 1          y = x + b    shows b = 4

so the line is y = x + 4

where the circle and the line intersect will be the max   x+y

x+4 = y

sub into the circle equation and solve for x = 6.47124      then y = 10.47124

Mx value of  (x+y)   is 16.94428

Here is a graph:

Jan 14, 2024

#1
+36919
+1

Find max of   x+y    start with standard circle form

x^2 -4x    + y^2 - 12y   = 0

(x-2)^2    + (y-6)^2   =  4 + 36

(x-2)^2   + (y-6)^2 =  40

Now find the 45 degree line that goes through the center   (2,6)

slope    = 1          y = x + b    shows b = 4

so the line is y = x + 4

where the circle and the line intersect will be the max   x+y

x+4 = y

sub into the circle equation and solve for x = 6.47124      then y = 10.47124

Mx value of  (x+y)   is 16.94428

Here is a graph:

ElectricPavlov Jan 14, 2024
#2
+128732
0

Nice, EP   !!!!

CPhill  Jan 14, 2024
#3
+36919
0

Thanx, Chris !

~EP

.....as an edit...the question asked for radicals in the answer

x = 2 + 2 sqrt(5)     y = 6 + 2 sqrt(5)

ElectricPavlov  Jan 14, 2024