First slide

Theory of expressions

Question

Given $x, y \in R, x^{2} + y^{2} > 0$ . Then the range of $\frac{x^{2} + y^{2}}{x^{2} + xy + 4 y^{2}}$ is

Moderate

A

$(\frac{10 - 4 \sqrt{5}}{30}, \frac{10 + 4 \sqrt{5}}{30})$
B

$(\frac{10 - 4 \sqrt{5}}{15}, \frac{10 + 4 \sqrt{5}}{15})$
C

$(\frac{5 - 4 \sqrt{5}}{15}, \frac{5 + 4 \sqrt{5}}{15})$
D

$(\frac{20 - 4 \sqrt{5}}{15}, \frac{20 + 4 \sqrt{5}}{15})$

Solution

$\begin{matrix} λ = \frac{x^{2} + y^{2}}{x^{2} + xy + 4 y^{2}} \\ = \frac{1 + {(\frac{y}{x})}^{2}}{1 + \frac{y}{x} + 4 {(\frac{y}{x})}^{2}} \\ = \frac{1 + z^{2}}{1 + z + 4 z^{2}} (Putting y / x = z) \end{matrix}$
$\begin{array}{l} or λ + λz + 4 {λz}^{2} = 1 + z^{2} \\ z^{2} (4 λ - 1) + z (λ) + λ - 1 = 0 \end{array}$
Since z is real,
$\begin{array}{ll} D \geq 0 \\ ∴ λ^{2} - 4 (4 λ^{2} - 5 λ + 1) \geq 0 \\ ∴ 15 λ^{2} - 20 λ + 4 \leq 0 \\ ∴ λ \in (\frac{10 - 4 \sqrt{5}}{15}, \frac{10 + 4 \sqrt{5}}{15}) \end{array}$

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