Question

The transformed equation of $x^2+y^2=r^2$ when the axes are rotated through an angle 36° is

• A. $\sqrt{5}X^{2}-4XY+Y^{2}=r^{2}$
• B. $X^{2}+2XY-\sqrt{5}Y^{2}=r^{2}$
• C. $X^{2}-Y^{2}=r^{2}$
• D. $X^{2}+Y^{2}=r^{2}$

Hint:

The equation of a circle centered at the origin ($x^2 + y^2 = r^2$) is invariant under rotation of axes.

Answer 1

The Correct Option is D

Step 1: Transformation Equations
For rotation of axes by angle $\theta$:
$x = X \cos \theta - Y \sin \theta$
$y = X \sin \theta + Y \cos \theta$
Step 2: Substitution
Substitute into $x^2 + y^2$:
$(X \cos \theta - Y \sin \theta)^2 + (X \sin \theta + Y \cos \theta)^2 = r^2$.
Step 3: Expansion
$X^2 \cos^2 \theta + Y^2 \sin^2 \theta - 2XY \sin \theta \cos \theta + X^2 \sin^2 \theta + Y^2 \cos^2 \theta + 2XY \sin \theta \cos \theta = r^2$.
Step 4: Conclusion
$X^2(\cos^2 \theta + \sin^2 \theta) + Y^2(\sin^2 \theta + \cos^2 \theta) = r^2$.
Since $\sin^2 \theta + \cos^2 \theta = 1$, the equation remains $X^2 + Y^2 = r^2$.
Final Answer: (d)