Question

If the pair of lines $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ intersect on $y$ axis then

• A. $2fgh=b{g}^2+c{h}^2$
• B. $b{g}^2\ne c{h}^2$
• C. $abc=2fgh$
• D. None of these

Answer 1

The correct option is A $2fgh=b{g}^2+c{h}^2$

As $s=a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represent a pair of line $\therefore \left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$
or $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=\mathrm{0....}\left(1\right)$ Now say point ofintersection onY axis be $(0,y_1$ and point of intersection of pair of line be obtained by solving the equations $\frac{\partial s}{\partial x}=0=\frac{\partial s}{\partial y}$ $\therefore \frac{\partial s}{\partial x}=0\Rightarrow ax+by+g=0$ $\begin{array}{c}\Rightarrow \\ \Rightarrow \end{array}\{\begin{array}{c}h{y}_1+g=0\\ b{y}_1+f=0\end{array}>(\ast )$ and $\frac{\partial s}{\partial y}=0\Rightarrow bx+by+f=0$ On compairing the equation given in (*) we get $bg=fh$ and $b{g}^2=fgh....\left(2\right)$ Again $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ meet at y-axis $\therefore x=0$ $\Rightarrow b{y}^2+2fy+c=0$ whose roots must be equal $\Rightarrow b{y}^2+2fy+c=0$ whose roots must be equal $\therefore f^2=bca{f}^2=abc......\left(3\right)$ Now using (2) and (3) in equation (I) we have $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$ $\Rightarrow (abc-a{f}^2)+(fgh-b{g}^2)+fgh-c{h}^2=0$ $\Rightarrow 0+0+fgh-c{h}^2=0\therefore c{h}^2=fgh.....\left(4\right)$ Now adding (2) and (4) $2fgh=c{h}^2+b{g}^2$