If-x-y-R-satisfy-in-equation-x-4-y-2-x-y-find-range-of-x-

Question Number 82719 by john santu last updated on 23/Feb/20

If x, y \in R s a t i s f y i n e q u a t i o n

x - 4 \sqrt{y} = 2 \sqrt{x - y} . f i n d r a n g e o f x

Commented by MJS last updated on 23/Feb/20

4 ⩽ x ⩽ 20

I will show later

for 4 ⩽ x < 16 we get 1 solution for y

for 16 ⩽ x ⩽ 20 we get 2 solutions for y

Commented by TANMAY PANACEA last updated on 23/Feb/20

t h a n k y o u s i r \dots

Commented by john santu last updated on 24/Feb/20

t h a n k y o u a l l

Answered by TANMAY PANACEA last updated on 23/Feb/20

1) x ⩾ y

2) y ⩾ 0

★ n o w i f y = 0

x = 2 \sqrt{x}

\sqrt{x} (\sqrt{x} - 2) = 0

s o e i t h e r x = 0 o r x = 4

w h e n x = 0 t h e n 0 y = 0

★ w h e n x = y

x - 4 \sqrt{x} = 0

\sqrt{x} (\sqrt{x} - 4) = 0

x = 0 o r x = 16

w h e n x = 0 t h e n y = 0

l e t o t h e r s t h r o w l i g h t i n [t h i s p r o b l e m

Answered by mr W last updated on 23/Feb/20

y ⩾ 0

x ⩾ y ⩾ 0

x ⩾ 4 \sqrt{y}

x^{2} - 8 x \sqrt{y} + 16 y = 4 (x - y)

20 y - 8 x \sqrt{y} + x^{2} - 4 x = 0

\sqrt{y} = \frac{2 x \pm \sqrt{x (20 - x)}}{10} ⩾ 0

x (20 - x) ⩾ 0

\Rightarrow 0 ⩽ x ⩽ 20 \dots (i)

2 x + \sqrt{x (20 - x)} ⩾ 0 \Rightarrow a l w a y s t r u e

2 x - \sqrt{x (20 - x)} ⩾ 0

4 x^{2} ⩾ 20 x - x^{2}

x^{2} ⩾ 4 x

\Rightarrow x ⩾ 4 \dots (i i)

f r o m (i) a n d (i i) w e g e t t h e r a n g e o f x :

4 ⩽ x ⩽ 20

x^{2} - 4 (1 + 2 \sqrt{y}) x + 20 y = 0

x = 2 (1 + 2 \sqrt{y}) \pm 2 \sqrt{1 + 4 \sqrt{y} - y}

1 + 4 \sqrt{y} - y ⩾ 0

5 - {(2 - \sqrt{y})}^{2} ⩾ 0

∣ 2 - \sqrt{y} ∣⩽ \sqrt{5}

- \sqrt{5} ⩽ 2 - \sqrt{y} ⩽ \sqrt{5}

\sqrt{y} ⩽ 2 + \sqrt{5} \Rightarrow y ⩽ {(2 + \sqrt{5})}^{2} = 9 + 4 \sqrt{5}

\sqrt{y} ⩾ 2 - \sqrt{5} \Rightarrow a l w a y s t r u e w i t h y ⩾ 0

t h e r a n g e o f y i s :

0 ⩽ y ⩽ 9 + 4 \sqrt{5}

Commented by TANMAY PANACEA last updated on 23/Feb/20

t h a n k y o u s i r \dots