If-x-y-R-x-2-xy-4-0-ax-2-b-y-2-x-4a-0-Find-the-minimum-of-a-2-1-2-b-2-

Question Number 108171 by ZiYangLee last updated on 15/Aug/20

If x, y \in R

x^{2} - x y + 4 = 0

{a x}^{2} + (b + y^{2}) x + 4 a = 0

Find the minimum of a^{2} + \frac{1}{2} b^{2}

Answered by mr W last updated on 15/Aug/20

c a s e 1 : a = 0

f r o m (i i) :

(b + y^{2}) x = 0

\Rightarrow x = 0 \Rightarrow i m p o s s i b l e, o t h e r w i s e 4 = 0!

\Rightarrow b + y^{2} = 0

f r o m (i) :

Δ = y^{2} - 4 \times 4 ⩾ 0

- b - 16 ⩾ 0

b ⩽ - 16

a^{2} + \frac{b^{2}}{2} = \frac{b^{2}}{2} ⩾ \frac{16^{2}}{2} = 128

c a s e 2 : a \neq 0

Δ = y^{2} - 4 \times 4 ⩾ 0 \Rightarrow y^{2} ⩾ 16 \Rightarrow y ⩽ - 4 o r ⩾ 4

{a x}^{2} - a x y + 4 a = 0 \dots (i i i)

(i i) - (i i i) :

(b + y^{2}) x + a x y = 0

x (y^{2} + a y + b) = 0

\Rightarrow x = 0 \Rightarrow i m p o s s i b l e, o t b e r w i s e 4 = 0!

\Rightarrow y^{2} + a y + b = 0

Δ = a^{2} - 4 b ⩾ 0

y_{1} = \frac{- a - \sqrt{Δ}}{2}, y_{2} = \frac{- a + \sqrt{Δ}}{2}

c a s e 2.1 : \sqrt{}

y_{2} = \frac{- a + \sqrt{Δ}}{2} ⩽ - 4

- a + \sqrt{Δ} ⩽ - 8

a ⩾ 8 + \sqrt{Δ}

\Rightarrow a ⩾ 8 a n d b ⩾ 16

\Rightarrow a^{2} + \frac{b^{2}}{2} ⩾ 192

c a s e 2.2 : \sqrt{}

y_{1} = \frac{- a - \sqrt{Δ}}{2} ⩾ 4

a \sqrt{} ⩽ - 8 - \sqrt{Δ}

\Rightarrow a ⩽ - 8 a n d b ⩽ - 16

\Rightarrow a^{2} + \frac{b^{2}}{2} ⩾ 192

c a s e 2.3 : \sqrt{}

y_{1} = \frac{- a - \sqrt{Δ}}{2} ⩽ - 4

a ⩾ 8 - \sqrt{Δ}

y_{2} = \frac{- a + \sqrt{Δ}}{2} ⩾ 4

a ⩽ - (8 - \sqrt{Δ})

\Rightarrow a = 0, b = - 16

Commented by mr W last updated on 15/Aug/20

a n s w e r i s 128

Commented by ZiYangLee last updated on 15/Aug/20

answer is \frac{128}{9}