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. $2fgh=af+c{h}^2$

Answer 1

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

Solution: $a{x}^2+b{y}^2+2gx+2fy+2hxy+c=0$ above equation represents pair of straight lines if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$\therefore$

pair of lines intersect on y axis $x=0$

$b{y}^2+2fy+c=0$

it is a quadratic in 'y'. so to get point of intersection on y axis we need two equal and real roots of this quadratic

$\therefore D=0$

$\Rightarrow 4f^2-4bc=0$

$\Rightarrow f^2=bc$

put this value in equation

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

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

Answer : option (A)