Assume that x and y are real numbers such that :

(x + 5)^2 + (y - 12)^2 = 196

What is the maximum value of the quantity x^2 + y^2 ?

Thank you very much for your help!! I appreciate it!

2. Originally Posted by rlarach
Assume that x and y are real numbers such that :

(x + 5)^2 + (y - 12)^2 = 196

What is the maximum value of the quantity x^2 + y^2 ?

Thank you very much for your help!! I appreciate it!
Not that I've bothered working the details (becuase for one thing I don't know your mathematical background), but the method of Lagrange multipliers comes to mind as a simple general approach to try .....

But no doubt there's another - tricky fancy - way.

3. Originally Posted by mr fantastic
Not that I've bothered working the details (becuase for one thing I don't know your mathematical background), but the method of Lagrange multipliers comes to mind as a simple general approach to try .....

But no doubt there's another - tricky fancy - way.
you know i have no experience with Lagrange multipliers

i was thinking of doing something along the lines of expanding the left hand side and solving for x^2 + y^2. so you would have x^2 + y^2 = f(x,y), where f(x,y) is, of course, a function of x and y. then you would proceed to minimize that function. but that seems like calc 3 stuff. and maybe is too much work anyway

4. What I tried doing was to get the center of this circle (assuming it is one) to be (-5,12) an the radius 14. Then I took each extrem (up down left and right ) and I did 26 squared plus 25 to get 701. But I'm not sure if there are other coordinates on the circle with a greater value when the squares are summed.

5. Originally Posted by rlarach
What I tried doing was to get the center of this circle (assuming it is one) to be (-5,12) an the radius 14. Then I took each extrem (up down left and right ) and I did 26 squared plus 25 to get 701. But I'm not sure if there are other coordinates on the circle with a greater value when the squares are summed.
Good to see that you've had a crack at this problem.

Originally Posted by mr fantastic
Not that I've bothered working the details (becuase for one thing I don't know your mathematical background), but the method of Lagrange multipliers comes to mind as a simple general approach to try .....

But no doubt there's another - tricky fancy - way.
I think there's a nice way to solve it - in fact, your working gave me the idea ......

Originally Posted by rlarach
What I tried doing was to get the center of this circle (assuming it is one) to be (-5,12) an the radius 14.
This is correct. What you now want is the circle C of the form x^2 + y^2 = r^2 that intersects the circle (x + 5)^2 + (y - 12)^2 = 196 and has the largest possible value of r.

If you think about it, you'll see that by symmetry the intersection of the line passing through (0, 0) and (-5, 12) with the circle (x + 5)^2 + (y - 12)^2 = 196 will give you the coordinates of a point on C. Use that point to get the value of r^2 and hence x^2 + y^2.

I think that should work. The details are left for you.