Question:
The normal to the curve $x = 9(1 + \cos \theta), y = 9 \sin \theta$ at $\theta$ always passes through the fixed point}
The normal to the curve $x = 9(1 + \cos \theta), y = 9 \sin \theta$ at $\theta$ always passes through the fixed point}
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The normal to a circle is always a radial line passing through its center.
Updated On: Apr 26, 2026
- (9, 0)
- (8, 9)
- (0, 9)
- (9, 8)
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The Correct Option is A
Solution and Explanation
Step 1: Identify Curve
$(x-9) = 9 \cos \theta$ and $y = 9 \sin \theta$.
$(x-9)^2 + y^2 = 81 \cos^2 \theta + 81 \sin^2 \theta = 81$.
This is a circle with center $(9, 0)$ and radius 9.
Step 2: Geometric Property
The normal to any point on a circle always passes through the center of that circle.
Step 3: Conclusion
The center is $(9, 0)$. Therefore, the normal always passes through $(9, 0)$.
Final Answer: (A)
$(x-9) = 9 \cos \theta$ and $y = 9 \sin \theta$.
$(x-9)^2 + y^2 = 81 \cos^2 \theta + 81 \sin^2 \theta = 81$.
This is a circle with center $(9, 0)$ and radius 9.
Step 2: Geometric Property
The normal to any point on a circle always passes through the center of that circle.
Step 3: Conclusion
The center is $(9, 0)$. Therefore, the normal always passes through $(9, 0)$.
Final Answer: (A)
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