Question

Find k, if the sum of slopes of the lines represented by the equation `x^(2) + kxy - 3y^(2) = 0` I s twice their product.

Answer 1

Let the lines represented by the equation.
` x^(2) + kxy - 3y^(2) = 0`
or , ` - x^(2)/3 - k/3 xy + y^(2) = 0 " be " y = m_(1) x and y = m_(2) x ,`
where `m_(1) and m_(2)` are the slopes of the lines.
` :. " " - x^(2)/3 - k/3 xy + y^(2) = (y - m_(1) x) (y - m_(2) x)`
or ` - x^(2)/3 - k/3 xy + y^(2) = y^(2) - (m_(1) + m_(2)) xy`
Comparing both sides, we get
` (m_(1) + m_(2)) = k/3 and , (m_(1) m_(2)) = - 1/3`
Given, ` m_(1) + m_(2) = 2 m_(1) m_(2)`
` :. " " k/3 = 2 xx (-1/3)`
or ` k = - 2 `