The transformed equation of x2+y2=r2 when the axes are rotated through an angle 36° is

Question

The transformed equation of x2+y2=r2 when the axes are rotated through an angle 36° is (A) √5 X2-4 X Y+Y2=r2 (B) X2+2 X Y-√5 Y2=r2 (C) X2-Y2=r2 (D) X2+Y2=r2

Tardigrade Admin · Accepted Answer

Given equation is x2+y2=r2 . After rotation x=X cos 36°-Y sin 36° and y=X sin 36o+Y cos 36° ∴ X2( cos 2 36o+ sin 2 36o)+Y2( sin 2 36o+ cos 2 36o)=r2 ⇒ X2+Y2=r2

Q. The transformed equation of $x^{2} + y^{2} = r^{2}$ when the axes are rotated through an angle $3 6^{\circ}$ is

$5 X^{2} - 4 X Y + Y^{2} = r^{2}$

$X^{2} + 2 X Y - 5 Y^{2} = r^{2}$

$X^{2} - Y^{2} = r^{2}$

$X^{2} + Y^{2} = r^{2}$

Given equation is $x^{2} + y^{2} = r^{2} .$ After rotation
$x = X cos 3 6^{\circ} - Y sin 3 6^{\circ}$
and $y = X sin 3 6^{o} + Y cos 3 6^{\circ}$
$∴ X^{2} (cos^{2} 3 6^{o} + sin^{2} 3 6^{o}) + Y^{2} (sin^{2} 3 6^{o} + cos^{2} 3 6^{o}) = r^{2}$
$\Rightarrow X^{2} + Y^{2} = r^{2}$

Q. The transformed equation of x2+y2=r2 when the axes are rotated through an angle 36∘ is

5​X2−4XY+Y2=r2

X2+2XY−5​Y2=r2

X2−Y2=r2

X2+Y2=r2