Question

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

2x² + 7xy + 3y² = 0

Answer 1

Comparing the equation

2x² + 7xy + 3y² = 0 with

ax² + 2hxy + by² = 0, we get,

a = 2, 2h = 7 i.e. h = ${\displaystyle \frac{7}{2}}$ and b = 3

Let θ be the acute angle between the lines.

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

= ${\displaystyle \left|\frac{2\sqrt{{\left(\frac{7}{2}\right)}^2-2\left(3\right)}}{2+3}\right|}$

= ${\displaystyle \left|\frac{2\sqrt{\left(\frac{49}{4}\right)-6}}{5}\right|}$

= ${\displaystyle \left|\frac{2\sqrt{\left(\frac{49-24}{4}\right)}}{5}\right|}$

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

= ${\displaystyle \frac{2\times \left(\frac{5}{2}\right)}{5}}$

= ${\displaystyle \frac{5}{5}}$

tan θ = 1

∴ θ = tan 1 = 45°

∴ θ = 45°