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