Question

The condition that the equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$, can take the form $a{X}^2+2hXY+b{Y}^2=0$ by translating the origin to a suitable point is

• A. $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$
• B. $2fgh-a{f}^2-b{g}^2-c{h}^2=0$
• C. $abc-a{f}^2-b{g}^2-c{h}^2=0$
• D. $abc+2fgh=0$

Answer 1

The correct option is A $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$x=\frac{-2(hy+g)\pm \sqrt{4(hy+g{)}^2-4a(b{y}^2+2fy+c}}{2}$
For a linear relation between $x$ and $y$,

$D=0$
$\therefore 4(hy+g{)}^2-4a(b{y}^2+2fy+c)=0$
$y^2(h^2-ab)+2(hg-af)y+g^2-ac=0$
Now we have quadratic in $y$
The roots are equal,

$\therefore D=0$
$4(hg-af{)}^2-4(h^2-ab\left)\right(g^2-ac)=0$
$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$