# 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 $\left(lx+my\right)\left(nx+oy\right)=0$ 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 $\mathrm{tan}\left(2\theta \right)=\frac{2\sqrt{{h}^{2}-ab}}{a+b}$.
• Midpoint of Intercepts: The midpoint of intercepts made by the pair of lines on the coordinate axes is $\left(\frac{-h}{a},\frac{-h}{b}\right)$.
• 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 $tan\left(2\theta \right)=\frac{2\text{√}\left({B}^{2}-AC\right)}{A-C}$.
• 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 $\left(-\frac{2Bc-2Ea}{4AC-{B}^{2}},-\frac{2Ac-2Dc}{4AC-{B}^{2}}\right)$.
• 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.