Pair of Straight Lines

Pair of Straight Lines

1. General Equation of Second Degree:

  • Any equation of the form ax2+2hxy+by2+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 ax2+2hxy+by2=0 is of great significance.
  • The condition for such an equation to represent a pair of straight lines is abh2=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 ax2+2hxy+by2=0 when abh2=0.
  • These lines are represented by ax2+2hxy+by2=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 ax2+2hxy+by2=0 is given by tan(2θ)=2h2aba+b.
  • Midpoint of Intercepts: The midpoint of intercepts made by the pair of lines on the coordinate axes is (ha,hb).
  • Conditions for Parallel Lines and Perpendicular Lines: If h2=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 ax2+2hxy+by2=0, the equation ax22hxy+by2=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 ax2+2hxy+by2=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:
    • B2AC>0 (The equation represents a pair of distinct lines)
    • B2AC=0 (The equation represents a pair of coincident lines)
    • B2AC<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: x2a2+y2b2=1 or x2a2y2b2=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(2θ)=2(B2AC)AC.
  • Midpoint of Intercepts: The midpoint of the intercepts on any line ax+by+c=0 by the curve Ax2+2Bxy+Cy2=0 is (2Bc2Ea4ACB2,2Ac2Dc4ACB2).
  • Condition for Orthogonality: Two lines represented by Ax2+2Bxy+Cy2=0 and Ax2+2Bxy+Cy2=0 are perpendicular if AA+BB=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, B2AC=0 (for Ax² + 2Bxy + Cy² = 0).
  • Pair of Coincident Lines: When the lines coincide, B2AC=0 and AD2+2BEFCD2AE2BF2=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.