**Joint Equation of A Pair of Straight Lines**

The joint equation of the straight lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is (a₁x + b₁y + c₁) (a₂x + b₂y + c₂) = 0

a₁a₂x² + xy (a₁b₂ + a₂b₁) + b₁b₂y² + (a₁c₂ + a₂c₁) x + (b₁c₂ + b₂c₁) y + c₁c₂ = 0

Or ax² + 2hxy + by² + 2gx + 2fy + c = 0

Where a = a₁a₂, b = b₁b₂, 2h = a₁b₂ + a₂b₁, 2g = a₁c₂ + a₂c₁, 2f = b₁c₂ + b₂c₁ and c = c₁c₂

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

Thus, the joint equation of a pair of straight lines is general equation of second degree.

The joint equation of a pair of straight lines a₁x + b₁y = 0 and a₂x + b₂y = 0 is (a₁x + b₁y) (a₂x + b₂y) = 0

Or a₁a₂x² + b₁b₂y² + xy (a₁b₂ + a₂b₁) = 0

Or ax² + 2hxy + by² = 0 where a = a₁a₂, b = b₁b₂, 2h = a₁b₂ + a₂b₁,

The equation ax² + 2hxy + by² = 0 is known as the homogeneous equation of second degree.

Thus, the joint equation of a pair of straight lines passing through the origin is the homogeneous equation of second degree.

**Pair of Straight Lines through the Origin**

**Homogenous Equation of Second Degree: **A rational, integral, algebraic equation in two variables x and y is said to be a homogenous equation of the second degree, if the sum of the indices (exponents) of x and y in each term is equal to 2.

The general form of the homogenous equation of the second degree in x and y is ax² + 2hxy + by² = 0 where a, b and h are constants.

- The homogenous equation of second degree ax² + 2hxy + by² = 0 represents a pair of straight lines passing through the origin.
- The homogeneous equation of second degree ax² + 2hxy + by² = 0 represents a pair of straight lines passing through the origin. These lines are real and distinct, if h² > ab and coincident if h² = ab and the lines are imaginary i.e., they do not exist if h² < ab.
- If y = m₁ x and y = m₂ x are lines represented by the equation ax² + 2hxy + by² = 0 then m₁ + m₂ \(=\frac{-2h}{b}=-\frac{coeff\,\,of\,\,xy}{coeff\,\,of\,\,{{y}^{2}}}\).

And m₁m₂ \(=\frac{a}{b}=\frac{coeff\,\,of\,\,{{x}^{2}}}{coeff\,\,of\,\,{{y}^{2}}}\).

- The condition that pairs of lines ax² + 2hxy + by² = 0 and a’x² + 2h’xy + b’y² = 0 should have one line in common is 4(h’b – hb’) (ha’ – h’a) = (ab’ – a’b)²
- The product of perpendiculars let fall from the point (x₁, y₁) upon the pair of lines ax² + 2hxy + by² = 0 is \(\frac{ax_{1}^{2}+2h{{x}_{1}}{{y}_{1}}+by_{1}^{2}}{\sqrt{{{\left( a-b \right)}^{2}}+4{{h}^{2}}}}\).
- The centroid (x̄ , ȳ) of the triangle with sides ax² + 2hxy + by² = 0 and lx + my = 1 is given by \(\frac{\overline{x}}{bl-hm}=\frac{\overline{y}}{am-hl}=\frac{2}{3\left( a{{m}^{2}}-2hlm+b{{l}^{2}} \right)}\).
- The orthocenter of the triangle formed by the lines ax² + 2hxy + by² = 0 and lx + my = 1 is given by \(\frac{x}{l}=\frac{y}{m}=\frac{a+b}{a{{m}^{2}}-2hlm+b{{l}^{2}}}\).