Question

The transformed equation of $3x^2+3y^2+2xy=2$. when the coordinate axes are rotated through an angle of $45^{\circ }$, is

• A. $x^2+2y^2=1$
• B. $2x^2+y^2=1$
• C. $x^2+y^2=1$
• D. $x^2+3y^2=1$

Answer 1

Answer: (B)

When axes are rotated through $45^{\circ }$,
$x=x\cos \theta -y\sin \theta =\frac{x-y}{\sqrt{2}}$
$y=x\sin \theta -y\cos \theta =\frac{x+y}{\sqrt{2}}$
$3x^2+3y^2+2xy=2$
$\Rightarrow 3{\left(\frac{x-y}{\sqrt{2}}\right)}^2+3{\left(\frac{x+y}{\sqrt{2}}\right)}^2+2\left(\frac{x^2-y^2}{2}\right)=2$
$3\left(x^2+y^2-2xy\right)+3\left(x^2+y^2+2xy\right)+2x^2-2y^2=4$
$4\left(2x^2+y^2\right)=4$
$\Rightarrow 2x^2+y^2=1$