# Pair of Straight Lines

## Pair of Straight Lines

### 1. **General Equation of Second Degree:**

- Any equation of the form $a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$ represents a second-degree equation involving $x$ and $y$terms.

### 2. **Homogeneous Equation of Second Degree:**

- A homogeneous equation in the form $a{x}^{2}+2hxy+b{y}^{2}=0$ is of great significance.
- The condition for such an equation to represent a pair of straight lines is $ab-{h}^{2}=0$. When this holds true, the equation factors into two straight lines.

### 3. **Pair of Straight Lines:**

- A pair of straight lines can be expressed in the form $a{x}^{2}+2hxy+b{y}^{2}=0$ when $ab-{h}^{2}=0$.
- These lines are represented by $a{x}^{2}+2hxy+b{y}^{2}=0$ factorizing into $(lx+my)(nx+oy)=0$ where $l,m,n,o$ are constants.
- The equation of the pair of straight lines can also be written as $lx+my=0$ and $nx+oy=0$.

### 4. **Properties of Pair of Straight Lines:**

**Angle Between Pair of Lines:**The angle between the pair of straight lines given by $a{x}^{2}+2hxy+b{y}^{2}=0$ is given by $\mathrm{tan}(2\theta )=\frac{2\sqrt{{h}^{2}-ab}}{a+b}$.

**Midpoint of Intercepts:**The midpoint of intercepts made by the pair of lines on the coordinate axes is $(\frac{-h}{a},\frac{-h}{b})$.**Conditions for Parallel Lines and Perpendicular Lines:**If ${h}^{2}=ab$, the pair of lines will be perpendicular. If $h=0$ or $ab=0$, the pair of lines will be parallel.

### 5. **Conjugate Lines:**

- For a given pair of lines represented by $a{x}^{2}+2hxy+b{y}^{2}=0$, the equation $a{x}^{2}-2hxy+b{y}^{2}=0$ represents the conjugate lines.
- Conjugate lines share properties such as intersecting at right angles and having the same midpoint of intercepts.

### 6. **Point of Intersection of Lines:**

- To find the point of intersection of two lines, solve the simultaneous equations formed by the pair of lines.

### 7. **Slope of Lines:**

- The slope of the lines can be determined using the equations $a{x}^{2}+2hxy+b{y}^{2}=0$ by rearranging to $y=mx+c$form.

**8. Conditions for Pair of Straight Lines:**

- For a general equation Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0 to represent a pair of straight lines:
- ${B}^{2}-AC>0$ (The equation represents a pair of distinct lines)
- ${B}^{2}-AC=0$ (The equation represents a pair of coincident lines)
- ${B}^{2}-AC<0$ (The equation represents a pair of imaginary lines)

### 9. **Forms of Pair of Straight Lines:**

**General Form**: Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0**Standard Form**: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ or $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ (where a, b are real)**Slope Intercept Form**: y = mx ± c√(1 + m²) (where m is the slope)

**10. ****Properties and Relationships:**

**Angle between the Lines**: The angle θ between the lines represented by Ax² + 2Bxy + Cy² = 0 is given by $tan(2\theta )=\frac{2\text{\u221a}({B}^{2}-AC)}{A-C}$.**Midpoint of Intercepts**: The midpoint of the intercepts on any line $ax+by+c=0$ by the curve $A{x}^{2}+2Bxy+C{y}^{2}=0$ is $(-\frac{2Bc-2Ea}{4AC-{B}^{2}},-\frac{2Ac-2Dc}{4AC-{B}^{2}})$.**Condition for Orthogonality**: Two lines represented by $A{x}^{2}+2Bxy+C{y}^{2}=0$ and ${A}^{\mathrm{\prime}}{x}^{2}+2{B}^{\mathrm{\prime}}xy+{C}^{\mathrm{\prime}}{y}^{2}=0$ are perpendicular if $A{A}^{\mathrm{\prime}}+B{B}^{\mathrm{\prime}}=0$.

**11. ****Special Cases:**

**Pair of Perpendicular Lines**: When two lines are perpendicular, $A+C=0$ (for Ax² + 2Bxy + Cy² = 0).**Pair of Parallel Lines**: When the lines are parallel, ${B}^{2}-AC=0$ (for Ax² + 2Bxy + Cy² = 0).**Pair of Coincident Lines**: When the lines coincide, ${B}^{2}-AC=0$ and $A{D}^{2}+2BEF-C{D}^{2}-A{E}^{2}-B{F}^{2}=0$ (for Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0).

**12. Applications:**

- Pair of straight lines finds applications in various fields such as physics (electromagnetism, optics), engineering (mechanics, circuits), and computer graphics.