Given that x+y=13 and xy=24, find the distance from the point (x,y) to the origin.
Given equations x+y=13 and xy =24.
Solving first equation for y, we get
y = 13-x.
Substituting y=13-x in second equation, we get
x(13-x)= 24.
13x -x^2=24.
-x^2+13x =24.
-x^2+13x -24=0.
Dividing each term by -1, we get
x^2-13x+24=0.
Applying quadratic formula
x = (-b pm sqrt(b^2 - 4ac))/(2a)
a = 1, b = -13, c = 24
x = (13 + sqrt(73))/2
x = (13 - sqrt(73))/2
Plugging into x + y = 13
y = (13 - sqrt(73))/2
y = (13 + sqrt(73))/2
Then plugging in x^2 + y^2, we get x^2 + y^2 = ((13 + sqrt(73))/2)^2 + ((13 + sqrt(73))/2)^2 = 144
Therefore, x^2 + y^2 = 12
