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
\({{m}_{1}}=\frac{-h}{9}\),
\({{m}_{1}}^{2}=\frac{1}{12}\),
\({{\left( \frac{-h}{9} \right)}^{2}}=\frac{1}{12}\),
\(\frac{{{h}^{2}}}{81}=\frac{1}{12}\),
\({{h}^{2}}=\frac{81}{12}=\frac{27}{4}\),
\(h=\pm \sqrt{\frac{27}{4}}=\pm \frac{3\sqrt{3}}{2}\).