Question

The transformed equation of $3x^2+3y^2+2xy=2$. When the coordinate axes are rotated through an angle of $45$, 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)

$2x^2+y^2=1$

Since, the axes are rotated through an angle $45$, then we replace $(x,y)$ by
$(x\cos 45-y\sin 45,x\sin 45+y\cos 45)$
Given equation is $3x^2+3y^2+2xy=2$
Therefore, $3{(\frac{x}{\sqrt{2}}-\frac{y}{\sqrt{2}})}^2+3\frac{x+y}{\sqrt{2}}+2\frac{x-y}{\sqrt{2}}\frac{x+y}{\sqrt{2}}=2$
$\Rightarrow \frac{3}{2}(x^2+y^2+2xy)+\frac{3}{2}(x^2+y^2-2xy)+\frac{2}{2}(x^2-y^2)=2$
$\Rightarrow 4x^2=2y^2=2$
$\Rightarrow 2x^2+y^2=1$