# 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: $\mathrm{tan}\theta =\mid \frac{2\sqrt{\left(hpr-agq\right)\left(agq-bps\right)}}{\left(apr+bqs\right)}\mid$

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: $3{x}^{2}+4xy-4{y}^{2}+7x-6y-8=0$ and $2{x}^{2}-5xy-{y}^{2}+3x+2y-4=0$.

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: $\mathrm{tan}\theta =\mid \frac{2\sqrt{\left(hpr-agq\right)\left(agq-bps\right)}}{\left(apr+bqs\right)}\mid$ $\mathrm{tan}\theta =\mid \frac{2\sqrt{\left(\left(2\right)\left(2\right)\left(-4\right)-\left(3\right)\left(-5\right)\left(7\right)\right)\left(\left(-4\right)\left(-5\right)-\left(2\right)\left(3\right)\left(2\right)\right)}}{\left(3\cdot 2\cdot 2+\left(-4\right)\left(-5\right)\right)}\mid$ $\mathrm{tan}\theta =\mid \frac{2\sqrt{\left(16+105\right)\left(-20-12\right)}}{\left(12+20\right)}\mid$

$\mathrm{tan}\theta =\mid \frac{2\sqrt{121\cdot \left(-32\right)}}{32}\mid$

$\mathrm{tan}\theta =\mid \frac{22\cdot 4}{32}\mid$

$\mathrm{tan}\theta =\mid \frac{11}{4}\mid$

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 slope-intercept 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.