If the slope of one of the pairs of lines represented by equation `a^(3) x^(2) + 2hxy + b^(3) y^(2) = 0` is square of the other, then prove that
`ab(a+ b) = - 2h. `

Explanation

`a^(3) x^(2) + 2hxy + b^(3) y^(2)= 0`
`rArr b^(3) ((y)/(x))^(2) + 2h ((y)/(x)) + x^(3) = 0`
Lines are `y = mx, where m = m_(1) and m_(2).`
`rArr b^(3) m^(2) + 2hm + a^(3) = 0` (1)
Given that slope of one line is square of the other line.
So, roots are `alpha and alpha^(2)`.
`rArr alpha a//b,` which satisfies the equation (1)
So, `b^(3) (a^(2))/(b^(2)) + 2h (a)/(b) + a^(3) = 0`
`rArr ab^(2) + a^(2) b+ 2h = 0`
`rArr ab (a+b) = - 2h`