Question

Find the measure of the acute angle between the line represented by:

4x² + 5xy + y² = 0

Answer 1

Comparing the equation

4x² + 5xy + y² = 0 with

ax² + 2hxy + by² = 0, we get,

a = 4, 2h = 5 i.e. h = ${\displaystyle \frac{5}{2}}$ and b = 1

Let θ be the acute angle between the lines.

∴ tan θ = ${\displaystyle \left|\frac{2\sqrt{{\text{h}}^2-\text{ab}}}{\text{a + b}}\right|}$

= ${\displaystyle \left|\frac{2\sqrt{{\left(\frac{5}{2}\right)}^2-4\left(1\right)}}{4+1}\right|}$

= ${\displaystyle \left|\frac{2\sqrt{\left(\frac{25}{4}\right)-4}}{5}\right|}$

= ${\displaystyle \left|\frac{2\times \frac{3}{2}}{5}\right|}$

∴ tan θ = ${\displaystyle \frac{3}{5}}$

∴ θ = tan^-1 ${\displaystyle \left(\frac{3}{5}\right)}$