Question

The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi }{4}$ (in the anti clockwise direction) is $a{X}^2+2hXY+b{Y}^2=20$ then which of the following is/are correct

• A. $a+b=18$
• B. $h=14$
• C. $h=7$
• D. $a-b=12$

Answer 1

Answer: (A), (C), (D)

$a-b=12$

$\theta =\frac{\pi }{4}$, original equation is $x^2+6xy+8y^2=10$

$x=X\cos \theta -Y\sin \theta =\frac{X-Y}{\sqrt{2}}$

$y=X\sin \theta +Y\cos \theta =\frac{X+Y}{\sqrt{2}}$

$\therefore$ Transformed equation is ${\left(\frac{X-Y}{\sqrt{2}}\right)}^2+6\left(\frac{X-Y}{\sqrt{2}}\right)\left(\frac{X+Y}{\sqrt{2}}\right)+8{\left(\frac{X+Y}{\sqrt{2}}\right)}^2=10$

$\Rightarrow 15X^2+14XY+3Y^2=20$

$\Rightarrow a=15,2h=14,b=3$