Angle between Pair of Lines
Angle between Pair of Lines:
1. Equation of Pair of Lines:
 A pair of lines represented by $a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$ can be expressed as ${A}_{1}{x}^{2}+2{B}_{1}xy+{C}_{1}{y}^{2}+2{D}_{1}x+2{E}_{1}y+{F}_{1}=0$ and ${A}_{2}{x}^{2}+2{B}_{2}xy+{C}_{2}{y}^{2}+2{D}_{2}x+2{E}_{2}y+{F}_{2}=0$.
2. Angle between Pair of Lines Formula:
 The formula to find the angle between two lines $a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$ and $p{x}^{2}+2qxy+r{y}^{2}+2sx+2ty+u=0$ is given by: $\mathrm{tan}\theta =\mid \frac{2\sqrt{(hpragq)(agqbps)}}{(apr+bqs)}\mid $
where:

 $a,b,h$ are coefficients of the first line
 $p,r,q$ are coefficients of the second line
 $g,f,s,t$ are other coefficients
Example:
Given two lines: $3{x}^{2}+4xy4{y}^{2}+7x6y8=0$ and $2{x}^{2}5xy{y}^{2}+3x+2y4=0$.
Find the angle between these lines.
Solution:
Comparing the given equations with the standard form of $A{x}^{2}+2Bxy+C{y}^{2}+2Dx+2Ey+F=0$:
 For the first equation: ${A}_{1}=3$, ${B}_{1}=2$, ${C}_{1}=4$, ${D}_{1}=7$, ${E}_{1}=6$
 For the second equation: ${A}_{2}=2$, ${B}_{2}=5$, ${C}_{2}=1$, ${D}_{2}=3$, ${E}_{2}=2$
Applying the formula for the angle between lines: $\mathrm{tan}\theta =\mid \frac{2\sqrt{(hpragq)(agqbps)}}{(apr+bqs)}\mid $ $\mathrm{tan}\theta =\mid \frac{2\sqrt{((2)(2)(4)(3)(5)(7))((4)(5)(2)(3)(2))}}{(3\cdot 2\cdot 2+(4)(5))}\mid $ $\mathrm{tan}\theta =\mid \frac{2\sqrt{(16+105)(2012)}}{(12+20)}\mid $
$\mathrm{tan}\theta =\mid \frac{2\sqrt{121\cdot (32)}}{32}\mid $
$\mathrm{tan}\theta =\mid \frac{22\cdot 4}{32}\mid $
$\mathrm{tan}\theta =\mid \frac{11}{4}\mid $
Taking inverse tangent to find the angle: $\theta ={\mathrm{tan}}^{1}\left(\frac{11}{4}\right)$ $\theta \approx 70.8{8}^{\circ}$
Therefore, the angle between the given lines is approximately $70.8{8}^{\circ}$.
3. Derivation of the Angle Formula:
 Obtained from the formula for the angle between two lines using the slope of the lines and the formula for the angle between two vectors.
4. Properties and Observations:
 The angle between the lines is invariant, irrespective of the way the lines are represented (e.g., in standard form or slopeintercept form).
 If two lines are perpendicular, then the angle between them is 90 degrees, and the product of their slopes is 1.
5. Special Cases:
 Parallel Lines: If two lines are parallel, the angle between them is either 0 degrees or 180 degrees (cosine of the angle will be 1 or 1).
 Perpendicular Lines: When two lines are perpendicular, the angle between them is 90 degrees.
6. Applications:
 Geometry: Useful in determining the orientation of lines and their intersections.
 Trigonometry and Mathematics: Understanding angles between vectors and lines.
 Physics and Engineering: Used in various calculations involving lines and their relationships.
7. Graphical Representations:
 The concept of the angle between lines can be visually understood by plotting lines on the Cartesian plane and observing their intersection angles.