4x? +y? +xy = (2x+ 1/4 *y)? + 15/ 16 * y?
Exchange elements again
manufacture
u = 2x + 1/4 *y
v = √ 15 /4 *y
2x+y = 2x+ 1/4 * y+3/4 * y = u+3/4 * 4/√ 15 * v = u+√( 3/5)* v
That is to find the circle C: u in UoV coordinate system? + v? The fixed point (m, n) on = 1, and the maximum value of m+√ (3/5) * n.
So point (m, n)
Is the solution of the following equation
u + √(3/5) *v = k
u? + v? = 1
K is the maximum value sought.
Substitute v
u? + v? =u? + ((k-u)/√(3/5))?
= u? +5/3 *(k? -2ku+ u? )
= 1
therefore
8/3 u? - 10/3 ku + 5/3 k? - 1 =0
There is only one solution to the maximum value u, and the straight line u+√(3/5) *v = k is tangent to the circle.
delta = ( 10/3 k)? -4 *8/3 *(5/3 k? - 1) =0
Re-simplification
5 k? = 8
k = +/- 2√ 10 / 5
Discard negative values
k = 2√ 10 / 5