Question

The locus of the point of intersection of tangents to the circle $x=a\cosθ, y=a\sinθ$ at the points whose parametric angles differ by $π/2$ is

• A. a straight line
• B. a circle
• C. a pair of straight line
• D. None of the above

Hint:

The locus of the point of intersection of tangents to the circle $x=a\cosθ, y=a\sinθ$ at the points whose parametric angles differ by $π/2$ is

Answer 1

The Correct Option is B

Step 1: Concept
The equation of the circle is $x^{2}+y^{2}=a^{2}$.
Step 2: Analysis
Tangents at $\theta$ and $\theta+\pi/2$ are $x\cos\theta + y\sin\theta = a$ and $-x\sin\theta + y\cos\theta = a$.
Step 3: Evaluation
Squaring and adding both tangent equations:
$(x\cos\theta + y\sin\theta)^{2} + (-x\sin\theta + y\cos\theta)^{2} = a^{2} + a^{2}$
$x^{2}(\cos^{2}\theta + \sin^{2}\theta) + y^{2}(\sin^{2}\theta + \cos^{2}\theta) = 2a^{2}$.
Step 4: Conclusion
This simplifies to $x^{2}+y^{2}=2a^{2}$, which represents a circle.
Final Answer: (b)