## Sunday, December 7, 2008

### General equation of second degree for a pair of straight lines.

The equation ax² +2hxy+by²+2gx+2fy+c = 0 is known as general equation of second degree.

The equation ax² +2hxy+by² = 0 is known as homogeneous equation of second degree.
In a homogeneous equation of second degree, the sum of indices (exponents) of x and y in each term is equal to 2.

The homogeneous equation of second degree ax² +2hxy+by² = 0 represents a joint equation of two straight lines passing through the origin if h²≥ab.

If y = m1x and y = m2x are the lines represented by a homogeneous equation of second degree ax² +2hxy+by² = 0, then

(i) m1 =m2 = -2h/b
(ii) m1m2 = a/b

The angle θ between the pair of lines represented the homogeneous equation of second degree ax² +2hxy+by² = 0 is given by

tan θ = [2√(h² –ab)]/(a+b)

If θ = 0, which means h² = ab lines are coincident.

Lines are perpendicular means θ = π/2, tan θ = ∞, and cot θ = 0.
This means a+b = 0 or a = -b
Coefficient of x² = coefficient of y²

ax² +2hxy+by²+2gx+2fy+c = 0 will represent a pair of straight lines if the determinant

|a h g|
|h b f|
|g f c|

= 0

Expanding the determinant

abc +2fgh -af² -bg² -ch² = 0

Angle θ between the lines represented by the general second degree equation ax² +2hxy+by²+2gx+2fy+c = 0 is given by

tan θ = [2 √(h² – ab)]/(a+b)

Algorithm to find separate equations of lines in ax² +2hxy+by²+2gx+2fy+c = 0

Step 1. Find factors for the homogeneous part ax² +2hxy+by². Let the factors be
(a1x +b1y ) and (a2x +b2y )

Step 2.Add constants c1 and c2 to them. (a1x +b1y +c1) and (a2x +b2y +c2).

Step 3. Multiply (a1x +b1y +c1) and (a2x +b2y +c2) and compare with ax² +2hxy+by²+2gx+2fy+c to obtain equations in c1 and c2.

Step 4. Solve the equations and get values of c1 and c2.