Question

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

• A. $2fgh=b{g}^2+c{h}^2$
• B. $b{g}^2\ne c{h}^2$
• C. $abc=2fgh$
• D. $Noneofthese$

Answer 1

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

Given pairs of lines

$S=a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0------\left(1\right)$ intersects at y-axis $x=0$

Let coordinate of point of intersection is $(0,y)$

To find the point of intersection:

$\frac{dS}{dx}=0$

$2ax+2hy+2g=0$

Here point of intersection is $(0,y)$ SO above eq passes through $(0,y)$

$0+2hy=-2g$

$y=-\frac{g}{h}$

So Intersection point $(0,-\frac{g}{h})$

Given pair passes through $(0,-\frac{g}{h})$

$b{(-\frac{g}{h})}^2+2f(-\frac{g}{h})+c=0$

$b{g}^2-2fgh+c{h}^2=0$

$2fgh=b{g}^2+c{h}^2$