Find the measure of acute angle between the lines represented by
$4 x^{2} + 5 x y + y^{2} = 0$

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Solution

Comparing the equation
$4 x^{2} + 5 x y + y^{2} = 0$ with
$a x^{2} + 2 h x y + b y^{2} = 0,$ we get,
$a = 4, 2 h = 5,$ i.e. $h = \frac{5}{2}$ and $b = 1.$
Let $θ$ be the acute angle between the lines.
$∴ tan θ = ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a + b} ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \frac{2 \sqrt{{(\frac{5}{2})}^{2} - 4 (1)}}{4 + 1} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \frac{2 \sqrt{(\frac{25}{4}) - 4}}{5} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \frac{2 \times \frac{3}{2}}{5} ∣ ∣ ∣ ∣ ∣$
$∴ tan θ = \frac{3}{5}$
$∴ θ = {tan}^{- 1} (\frac{3}{5})$

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Find the measure of acute angle between the lines represented by4x2+5xy+y2=0

Find the measure of acute angle between the lines represented by
$4 x^{2} + 5 x y + y^{2} = 0$