Question

find the angle through which the axes are to be rotated so as to remove the xy terms in equation x2 +4xy+y2-2x+2y-6=0
how to solve​

Answer 1

Transformation of co-ordinates

Solution: Let the axes be turned through an angle $\theta$.

Then, after removing the primes from $(x',\:y')$, the general form of the equation becomes

$\quad (x\:cos\theta-y\:sin\theta)^{2}-4\:(x\:cos\theta-y\:sin\theta)\:(x\:sin\theta+y\:cos\theta)+(x\:sin\theta+y\:cos\theta)^{2}-2\:(x\:cos\theta-y\:sin\theta)+2\:(x\:sin\theta+y\:cos\theta)-6=0$

The coefficient of $xy$ will be zero, if

$\quad -2\:sin\theta\:cos\theta+4\:cos^{2}\theta-4\:sin^{2}\theta+2\:sin\theta\:cos\theta=0$

$\Rightarrow 4\:(cos^{2}\theta-sin^{2}\theta)=0$

$\Rightarrow cos2\theta=0=cos\frac{\pi}{2}$

$\Rightarrow 2\theta=\frac{\pi}{2}$

$\Rightarrow \theta=\frac{\pi}{4}$

Answer: the required angle is $\frac{\pi}{4}$.

Answer 2

Answer:

Answer is π/4

Here is the solution.