Question

The equation of a curve is $3x^2+2\mathrm{xy}+3y^2=10$
. Its equation if the axes are rotated through an angle $45°$ will be .

• A. $2x^2+y^2=5$
• B. $2x^2+y^2=10$
• C. $x^2+2y^2=5$
• D. $x^2+2y^2=10$

Answer 1

The correct option is A $2x^2+y^2=5$

With the given conditions, we have
$x=\mathrm{Xcos}45°-\mathrm{Ysin}45°=\frac{X-Y}{\sqrt{2}}\mathrm{and}y=\mathrm{Xsin}45°+\mathrm{Ycos}45°=\frac{X+Y}{\sqrt{2}}\mathrm{Thus},\mathrm{the}\mathrm{equation}3x^2+2\mathrm{xy}+3y^2=10\mathrm{transforms}\mathrm{to}3(\frac{X-Y}{\sqrt{2}}{)}^2+2(\frac{X-Y}{\sqrt{2}}\left)\right(\frac{X+Y}{\sqrt{2}})+3(\frac{X+Y}{\sqrt{2}}{)}^2=10\mathrm{or}2X^2+Y^2=5\mathrm{Replacing}X\to x\mathrm{and}Y\to y2x^2+y^2=5$