Question

Transform to axes inclined at an angle $\alpha$ to the original axes the equations
(1) $x^2+y^2=r^2$,
and (2) $x^2+2xy\tan 2\alpha -y^2=a^2$.

Answer 1

(1) The value of $\theta =\alpha$

So the required equation after the transformation of axes are,

$X=xcos\alpha +ysin\alpha$

$Y=-xsin\alpha +ycos\alpha$$

And for the value of $x$ and $y$ in terms of $r$, we can substitute from $x^2+y^2=r^2$ by eliminating x and y respectively.

(2) We have $x^2+2xytan\alpha -y^2=a^2$

from here we the value of $\theta =\frac{\alpha }{2}$

So the desired equation of tranformed axes are,

$X=xcos\frac{\alpha }{2}+ysin\frac{\alpha }{2}$

$X=-xsin\frac{\alpha }{2}+ycos\frac{\alpha }{2}$