Technique explanation
The sutra Lopanasthapanabhyam ("By elimination and retention") is used to factorize second-degree equations into pairs of straight lines. A general second-degree equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of lines only if a specific determinant condition is met.
The method involves factorizing the homogeneous part, then isolating variables by setting x=0 and y=0 to find constant terms, and finally splicing the results. Standard fallback: Solving for y using the quadratic formula.
Worked examples
Example 1: Factorizing a Pair of Lines
Factorize x² - y² + 2x + 1 = 0.
1. Homogeneous part: x² - y² = (x-y)(x+y).
2. Set y=0: x² + 2x + 1 = (x+1)(x+1).
3. Set x=0: -y² + 1 = (1-y)(1+y).
4. Splice: (x - y + 1)(x + y + 1) = 0.
Check: Expand (x - y + 1)(x + y + 1) = x(x+y+1) - y(x+y+1) + 1(x+y+1) = x² - y² + 2x + 1. Verified.
Example 2: General Case
Factorize 12x² + 7xy - 10y² + 13x + 45y - 35 = 0.
1. Homogeneous: (3x - 2y)(4x + 5y).
2. Set y=0: 12x² + 13x - 35 = (3x + 7)(4x - 5).
3. Set x=0: -10y² + 45y - 35 = (-2y + 7)(5y - 5).
4. Splice: (3x - 2y + 7)(4x + 5y - 5) = 0.
Check: Expand (3x - 2y + 7)(4x + 5y - 5) = 12x² + 15xy - 15x - 8xy - 10y² + 10y + 28x + 35y - 35 = 12x² + 7xy - 10y² + 13x + 45y - 35. Verified.
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