Find the joint equation of the pair of lines through the origin which are perpendicular to the lines given by `5x^(2) + 2xy - 3y^(2) = 0.`

Explanation

Given equation is , ` 5x^(2) + 2xy - 3y^(2) = 0` ….(1)
Comparing with ` ax^(2) + 2hxy + by^(2) = 0, ` we get ` a = 5 , b = - 3, 2h = 2`
Let ` m_(1) and m_(2)` be the slopes of the lines represented by the given equation.
` :. " " m_(1) + m_(2) = (-2h)/b = (-2)/(-3) = 2/3`
` m_(1) * m_(2) = a/b = (-5)/3.`
Since the required lines are perpendicular to these lines, their slopes are ` - 1/m_(1) and 1/m_(2)`
Also, the required lines pass through the origin, therefore, their equations are
` y = - 1/m_(1) x`
and ` y = - 1/m_(2) x`
` :. " " x + m_(1) y = 0 and x = m_(2) y = 0 `
` :. ` Their joint equation is
` (x + m_(1) y) (x + m_(2) y) = 0`
` :. " " x^(2) + (m_(1) + m_(2)) xy + m_(1) m_(2) y^(2) = 0`
` :. " " x^(2) + 2/3 xy - 5/y y^(2) = 0`
` :. 3x^(2) + 2xy - 5y^(2) = 0` is the required equation.