Question

Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:

5x² + 2xy - 3y² = 0 

Answer 1

Comparing the equation 5x² + 2xy - 3y² = 0  with ax² + 2hxy + by² = 0, we get,

a = 5, 2h = 2, b= - 3

Let m₁ and m₂ be the slopes of the lines represented by 5x² + 2xy - 3y² = 0  

∴ `"m"_1 + "m"_2 = (-2"h")/"b" = (-2)/-3 = 2/3` and `"m"_1 "m"_2 = "a"/"b" = 5/-3`   ....(1)

Now required lines are perpendicular to these lines

∴ their slopes are `(-1)/"m"_1` and `(-1)/"m"_2`

Since these lines are passing through the origin, their separate equations are

y = `(-1)/"m"_1 "x"` and y = `(-1)/"m"_2 "x"`

i.e. m₁y = - x and m₂y = - x

i.e. x + m₁y = 0 and x + m₂y = 0

∴ their combined equation is

(x + m₁y)(x + m₂y) = 0

∴ x² + (m₁ + m₂)xy + m₁m₂y² = 0 

∴ `"x"^2 + 2/3 "xy" - 5/3 "y"^2 = 0`  ...[By (1)]

∴ 3x² + 2xy - 5y² = 0