Question

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

• A. $17x^2+16xy+17y^2=225$
• B. $17x^2+23y^2=391$
• C. $17x^2-16xy+17y^2=225$
• D. $23x^2+17y^2=391$

Answer 1

Answer: (C)

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
$9x^2+25y^2=225$
$\Rightarrow 9{\left(\frac{x-y}{\sqrt{2}}\right)}^2+25{\left(\frac{x+y}{\sqrt{2}}\right)}^2=225$
$\Rightarrow 9\left(\frac{x^2+y^2-2xy}{2}\right)+25\left(\frac{x^2+y^2+2xy}{2}\right)=225$
$\Rightarrow 9x^2+9y^2+18xy+25x^2+25y^2-50xy=450$
$\Rightarrow 34x^2+34y^2-32xy=450$
$\Rightarrow 17x^2+17y^2-16xy=225$