If the axes are rotated through an angle ' $\alpha$ ', then the number of values of $\alpha$ such that the…
If the axes are rotated through an angle ' $\alpha$ ', then the number of values of $\alpha$ such that the transformed equation of $x^2+y^2+2 x+2 y-5=0$ contains no linear terms is
$0$
$1$
$2$
infinite
Solution
$\begin{aligned}
& \quad x^2+y^2+2 x+2 y-5=0 \\
& \text { Let } x=\mathrm{X} \cos \alpha-\mathrm{Y} \sin \alpha \text { and } y=\mathrm{X} \sin \alpha+\mathrm{Y} \cos \alpha \\
& \therefore(\mathrm{X} \cos \alpha-\mathrm{Y} \sin \alpha)^2+(\mathrm{X} \sin \alpha+\mathrm{Y} \cos \alpha)^2 \\
& +2(\mathrm{X} \cos \alpha-\mathrm{Y} \sin \alpha)+2(\mathrm{X} \sin \alpha+\mathrm{Y} \cos \alpha)-5=0 \\
& \Rightarrow \mathrm{X}^2+\mathrm{Y}^2+2(\cos \alpha+\sin \alpha) \mathrm{X}+2(\cos \alpha-\sin \alpha) \mathrm{Y}-5=0
\end{aligned}$ Now, to remove linear terms
$\cos \alpha+\sin \alpha=0 \text { and } \cos \alpha-\sin \alpha=0$ Which is not possible for any $\alpha$.