Question

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$, if the equation $25 x^2 + 9y^2 = 225$ 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, $25 x^{2}+9 y^{2}=225$ becomes
$25\left(\frac{X-Y}{\sqrt{2}}\right)^{2}+9\left(\frac{X+Y}{\sqrt{2}}\right)^{2}=225$
$\Rightarrow 34 X^{2}+34 Y^{2}-32 X Y=450$
$\Rightarrow 17 X^{2}+17 Y^{2}-16 X Y=225$
On comparing, we get
$\alpha=\gamma=17, \beta=-16$ and $\delta=225$
$\therefore (\alpha+\beta+\gamma-\sqrt{\delta})^{2}$
$=(34-16-15)^{2}$
$=3^{2}=9$