How find the range of $4a+2b$

Question

let $a\ge 0,b\ge 0$,and such that

$$(a-2)^2+(b-1)^2=5$$

find the range of $4a+2b$

my try:

use Cauchy-Schwarz inequality

$$[(a-2)^2+(b-1)^2][16+4]\ge (4a-8+2b-2)^2$$ so $$-10\le 4a+2b-10\le 10$$ so $$0\le 4a+2b\le 20$$ But this problem anwser is $$4a+2b\in \{0\}\bigcup [4,20] $$

where is wrong? Thank you.and someone have other nice methods?

Answer 1

f(a,b)=4a+2b ========a^2+b^2
if you catch 4a+2b from the circle eqution so look at this idea