If the pair of lines `ax^2+2hxy+by^2+2gx+2fy+c=0` intersect on Y-axis , then

Options

A. `2fgh=bg^(2)+ch^(2)`
B. `bg^(2)+ch^(2)=fgh`
C. `abc=2fgh`
D. none of these

Answer

A

Explanation

The given equation represents a pair of straight lines.
`therefore" "abc+2fgh-af^(2)-bg^(2)-ch^(2)=0" …(i)"`
The coordinates of the point of intersection are obtained by solving
`ax+hy+g=0 and hx+by+f=0`
Since the point of intersection lies on y-axis. Therefore,
`hy+g=0 and by+f=0rArr-(g)/(h)=-(f)/(b)rArrbg=hf" ...(ii)"`
The y-coordiantes of the point of intersection of the lines with y-axis are the roots of the equation
`by^(2)+2fy+c=0`
This equation must have equal roots.
`therefore" "f^(2)=bc`