Question

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

• A. $X^2+2Y^2=1$
• B. $2X^2+Y^2=1$
• C. $X^2-2Y^2=1$
• D. $2X^2-Y^2=1$

Answer 1

The correct option is B $2X^2+Y^2=1$

Let $(X,Y)$ be the new coordinates of $(x,y)$ when the axes are rotated through an angle of ${45}^{\circ }$
Then $x=X\cos {45}^{\circ }-Y\sin {45}^{\circ }=\frac{X-Y}{\sqrt{2}},y=X\sin {45}^{\circ }+Y\cos {45}^{\circ }=\frac{X+Y}{\sqrt{2}}$
Hence the transformed equation is $3{\left(\frac{X-Y}{\sqrt{2}}\right)}^2+3{\left(\frac{X+Y}{\sqrt{2}}\right)}^2+2\left(\frac{X-Y}{\sqrt{2}}\right)\left(\frac{X+Y}{\sqrt{2}}\right)=2$
$\Rightarrow 8X^2+4Y^2=4$
$\Rightarrow 2X^2+Y^2=1$