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