# Angle between the Pair of Lines

## Angle between the Pair of Lines

An angle between the Pair of Line represented by ax² + 2hxy + by² = 0:

Let θ  be the angle between the lines. Then,

$$\tan \theta =\pm \frac{{{m}_{1}}-{{m}_{2}}}{1+{{m}_{1}}{{m}_{2}}}$$,

$${{\left( {{m}_{1}}-{{m}_{2}} \right)}^{2}}={{\left( {{m}_{1}}+{{m}_{2}} \right)}^{2}}-4{{m}_{1}}{{m}_{2}}$$,

$$=\pm \frac{\sqrt{{{\left( {{m}_{1}}+{{m}_{2}} \right)}^{2}}-4{{m}_{1}}{{m}_{2}}}}{1+{{m}_{1}}{{m}_{2}}}$$,

ax² + 2hxy + by² = 0

Sum of the roots is = -2h/b

m₁ + m₂ = -2h/b

Product of the roots is = a/b

m₁. m₂ = a/b

$$=\pm \frac{\sqrt{\frac{4{{h}^{2}}}{{{b}^{2}}}-4\frac{a}{b}}}{1+\left( \frac{a}{b} \right)}$$,

$$=\pm \frac{2\sqrt{\frac{{{h}^{2}}-ab}{{{b}^{2}}}}}{\left( \frac{a+b}{b} \right)}$$,

$$=\pm \frac{2\sqrt{{{h}^{2}}-ab}}{a+b}$$,

$$\tan \theta =\pm \frac{2\sqrt{{{h}^{2}}-ab}}{a+b}$$.

Note: If the lines are perpendicular, then θ = 90°. Therefore

$$\tan \theta =\pm \frac{2\sqrt{{{h}^{2}}-ab}}{a+b}$$

$$\cot \theta =0$$,

$$\frac{\sin \theta }{\cos \theta }=\pm \frac{2\sqrt{{{h}^{2}}-ab}}{a+b}$$,

$$\frac{\sin {{90}^{o}}}{\cos {{90}^{o}}}=\pm \frac{2\sqrt{{{h}^{2}}-ab}}{a+b}$$,

$$\frac{\sin {{90}^{o}}}{0}=\pm \frac{2\sqrt{{{h}^{2}}-ab}}{a+b}$$.

a + b = 0

(coeff. of x²) + (coeff. of y²) = 0

Example: Find the value of a for which the lines represented by ax² + 5xy + 2y² = 0 are mutually perpendicular.

Solution:

Mutually perpendicular (θ = 90°)

(coeff. of x²) + (coeff. of y²) = 0

a + b = 0

a + 2 = 0

a = -2