Q. If the equation of a curve $C$ is transformed to $9x^2 + 25y^2 = 225$ by the rotation of the coordinate axes about the origin through an angle $\frac{\pi}{4}$ in the positive direction then the equation of the curve $C$, before the transformation is
AP EAMCETAP EAMCET 2019
Solution:
Let the original and new coordinates of the curve are as following form
x
y
x
$\cos \frac{\pi}{4}$
$\sin \frac{\pi}{4}$
y
$- \sin \frac{\pi}{4}$
$\cos \frac{\pi}{4}$
Now, $x=\frac{x}{\sqrt{2}}-\frac{y}{\sqrt{2}} \dots$(i)
and $y=\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}} \dots$(ii)
Substituting Eqs. (i) and (ii) in curve
$ 9 x^{2}+25 y^{2}=225 $
$ \Rightarrow 9\left(\frac{x-y}{\sqrt{2}}\right)^{2}+25\left(\frac{x+y}{\sqrt{2}}\right)^{2}=225 $
$ \Rightarrow 9\left(\frac{x^{2}+y^{2}-2 x y}{2}\right)+25\left(\frac{x^{2}+y^{2}+2 x y}{2}\right)=225 $
$ \Rightarrow 9 x^{2}+9 y^{2}+18 x y+25 x^{2}+25 y^{2}-50 x y=450 $
$\Rightarrow 34 x^{2}+34 y^{2}-32 x y=450$
$ \Rightarrow 17 x^{2}+17 y^{2}-16 x y=225 $
| x | y | |
|---|---|---|
| x | $\cos \frac{\pi}{4}$ | $\sin \frac{\pi}{4}$ |
| y | $- \sin \frac{\pi}{4}$ | $\cos \frac{\pi}{4}$ |