Question

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi }{4}$, if the equation $25x^2+9y^2=225$ is transformed to $\alpha x^2+\beta xy+\gamma y^2=\delta$, then $(\alpha +\beta +\gamma -\sqrt{\delta }{)}^2$ =

• A. 3
• B. 9
• C. 4
• D. 16

Answer 1

Answer: (B)

After rotation of coordinate axes about the origin in the positive direction through on angle $\frac{\pi }{4}$, the new coordinates are $(X,Y)$ have relation with older coordinates $(x,y)$ is
$(x,y)=[(X\cos \theta -Y\sin \theta ),(Y\cos \theta +X\sin \theta ))$, where
$\theta =\frac{\pi }{4}$
$=\left(\left(\frac{X}{\sqrt{2}}-\frac{Y}{\sqrt{2}}\right),\left(\frac{Y}{\sqrt{2}}+\frac{X}{\sqrt{2}}\right)\right)$
so, $25x^2+9y^2=225$ becomes
$25{\left(\frac{X-Y}{\sqrt{2}}\right)}^2+9{\left(\frac{X+Y}{\sqrt{2}}\right)}^2=225$
$\Rightarrow 34X^2+34Y^2-32XY=450$
$\Rightarrow 17X^2+17Y^2-16XY=225$
On comparing, we get
$\alpha =\gamma =17,\beta =-16$ and $\delta =225$
$\therefore (\alpha +\beta +\gamma -\sqrt{\delta }{)}^2$
$=(34-16-15{)}^2$
$=3^2=9$