If the pair of lines `ax^(2)+2hxy+by^(2)+2gx+2fy+c=0` intersect on the y-axis , then prove that `2fgh=bg^(2)+ch^(2)`.

Explanation

Let the line intersect on y- axis at `p(0,y_(1))`.
Putting this point in the equation of straight lines , we get `by_(1)^(2)+2fy_(1)+c=0`
Above equation must have equal roots .
`:.4f^(2)-4bc=0`
or `f^(2)=bc`
Also , given equation represents pair of straight lines.
`:.abc+2fgh-af^(2)-bg^(2)-ch^(2)=0`
From (1) , putting the value of `f^(2)` in (2), we get
`2fgh=bg^(2)+ch^(2)`