Question

The transformed equation of $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ when the axes are rotated through an angle of ${90}^{\circ }$ is

• A. $b{X}^2-2hXY+a{Y}^2+2fX-2gY+c=0$
• B. $b{X}^2+2hXY+a{Y}^2+2fX+2gY+c=0$
• C. $b{X}^2-2hXY+a{Y}^2-2fX+2gY+c=0$
• D. $b{X}^2-2hXY+a{Y}^2-2fX-2gY+c=0$

Answer 1

The correct option is A $b{X}^2-2hXY+a{Y}^2+2fX-2gY+c=0$

$x=X\cos 90{}^{\circ }-Y\sin 90{}^{\circ }=-Yy=X\sin 90{}^{\circ }+Y\cos 90{}^{\circ }=X$
Hence the transformed equation is
$a(-Y{)}^2+2h(-Y\left)\right(X)+b{X}^2+2g(-Y)+2fX+c=0\Rightarrow b{X}^2-2hXY+a{Y}^2+2fX-2gY+c=0$