Pair of straight lines:

The general equation of 2nd degree ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents the pair of straight lines if   and h² ≥  ab.

=> abc + 2fgh - af² - bg² - ch² = 0 and h² ≥ ab.

The homogeneous 2nd degree equation ax² + 2hxy + by² = 0 represents the pair of straight lines through the origin if h² ≥ ab.

If lines through the origin whose joint equation is ax² + 2hxy + by² = 0, are y = m₁x and
y = m₂x, then  y² - (m₁ + m₂)xy + m₁m₂x² = 0 and  are identical, such that

.  

If q be the angle between 2 lines, through the origin, then

The lines given are perpendicular if a + b = 0 and coincident if h² = ab.

The joint Equation of Pair of Lines Joining the Origin and the Points of Intersection of the Curve and a Line:

Illustration: If lines joining origin to the points of intersection of curve 2x² + 3y² + m = 0 and the line y = 2x + 3 are perpendicular, and then find the value of m.

Solution.  Homogenise given curve with the line, we get the desired line

                      As, the lines are perpendicular

                      => Coefficient of x² + coefficient of y² = 0

                      => m = -9

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