The combined equation of the pair of lines through the origin and perpendicular to the pair of lines given by `ax^(2)+2hxy+by^(2)=0`, is

Explanation

(i) Let `m_(1)` and `m_(2)` be the slopes of lines represented by `ax^(2)+2hxy+by^(2)=0`.
`:.m_(1)+m_(2)=-(2h)/(b), m_(1)m_(2)=(a)/(b)` ...(1)
The required lines are perpendicular to given lines.
`:.` Their slopes are `-(1)/(m_(1))` and `(1)/(m_(2))`.
`:'` The required lines passing through origin ,
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`
`s:.x^(2)+(m_(1)+m_(2))xy+m_(1)m_(2)y^(2)=0`
`:.x^(2)+((-2h)/(b))xy+((a)/(b))y^(2)=0` [From eq. (1)]
`:.bx^(2)-2hxy+ay^(2)=0`
This is the required equation.