If-4-log-x-2y-4-log-x-2y-1-Minimum-value-of-x-y-is-

Question Number 161166 by naka3546 last updated on 13/Dec/21

I f^{4} \log (x + 2 y) +^{4} \log (x - 2 y) = 1 .

M i n i m u m v a l u e o f ∣ x ∣ - ∣ y ∣ i s \dots ?

Answered by mr W last updated on 13/Dec/21

(x + 2 y) (x - 2 y) = 4

{(\frac{x}{2})}^{2} - y^{2} = 1 \Rightarrow h y p e r b o l a

2 (\frac{x}{2}) \frac{1}{2} - 2 y \frac{d y}{d x} = 0

2 (\frac{x}{2}) \frac{1}{2} - 2 y \times 1 = 0

y = \frac{x}{4}

{(\frac{x}{2})}^{2} - {(\frac{x}{4})}^{2} = 1

\Rightarrow x = \pm \frac{4}{\sqrt{3}}

\Rightarrow y = \pm \frac{1}{\sqrt{3}}

∣ x ∣ - ∣ y ∣= \frac{4}{\sqrt{3}} - \frac{1}{\sqrt{3}} = \sqrt{3} = m i n i m u n

Commented by naka3546 last updated on 13/Dec/21

T h a n k y o u, s i r .