# Pair of Straight Lines -Problems

### Pair of Straight Lines -Problems

1. Find the acute angle between the pair of line represented by the following equation x² – 7xy + 12y² = 9.

Solution: Given that,

x² – 7xy + 12y² = 9

Comparing with ax² + 2hxy + by² = 0

a = 1,

b = 12 and

h = -7/2

Let θ be the angle between the lines, then tanθ $$=\frac{2\sqrt{{{h}^{2}}-ab}}{a+b}$$

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

tanθ  $$=\frac{2\sqrt{\frac{1}{4}}}{13}$$,

tanθ $$=\frac{\sqrt{1}}{13}$$,

$$\tan \theta =\left( \frac{1}{13} \right)$$,

$$\theta ={{\tan }^{-1}}\left( \frac{1}{13} \right)$$.

2.  Find the value of h, if the slope of the lines represented by 6x² + 2hxy + y² = 0 are in the ratio 1 : 2

Solution: Combined equation of the pair of lines is 6x² + 2hxy + y² = 0

Let y = m₁ x and y = m₂ x be the represented by 6x² + 2hxy + y² = 0

∴ m₁ + m₂ = -2h/6 = -h/3

m₁m₂ = 1/6

Given m₁ / m₂ = ½

⇒ 2m₁ = m₂

m₁ + 2m₁ = -h/3

3m₁ = -h/3

2m₁² = ⅙

$${{m}_{1}}=\frac{-h}{9}$$,

$${{m}_{1}}^{2}=\frac{1}{12}$$,

$${{\left( \frac{-h}{9} \right)}^{2}}=\frac{1}{12}$$,

$$\frac{{{h}^{2}}}{81}=\frac{1}{12}$$,

$${{h}^{2}}=\frac{81}{12}=\frac{27}{4}$$,

$$h=\pm \sqrt{\frac{27}{4}}=\pm \frac{3\sqrt{3}}{2}$$.