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Problem 13

# The equation of a circle $$C_{1}$$ is $$x^{2}+y^{2}=4$$. The locus of the intersection of orthogonal tangents to the circle is the curve $$C_{2}$$ and the locus of the intersection of perpendicular tangents to the curve $$C_{2}$$ is the curve $$C_{3}$$. Then : (a) $$C_{3}$$ is a circle (b) The area enclosed by the curve $$C_{3}$$ is $$8 \pi$$ (c) $$C_{2}$$ and $$C_{3}$$ are circles with the same centre (d) none of these

Expert verified
The correct answers are $$C_{3}$$ is a circle (option a), $$C_{2}$$ and $$C_{3}$$ are circles with the same centre (option c). The area contained by $$C_{3}$$ is not $$8\pi$$, it is $$4\pi$$, hence option b is incorrect, and option d is also incorrect.
See the step by step solution

## Step 1: Identifying Characteristics of Curve $$C_{1}$$

From the given circle equation $$x^{2}+y^{2}=4$$, we can identify that the center of the circle $$C_{1}$$ is at the origin (0,0) and the radius is $$r=2$$.

## Step 2: Find the Locus for the Intersection Points of Orthogonal Tangents to $$C_{1}$$

It is known in geometry that the locus of the intersection points of perpendicular tangents to a given circle forms another circle with the same center. Therefore, $$C_{2}$$ will be a circle with the same center as $$C_{1}$$ which is (0, 0).

## Step 3: Find the Locus for the Intersection Points of Orthogonal Tangents to $$C_{2}$$

Applying the same logic as in Step 2, the locus of the intersection points of the orthogonal tangents to the curve $$C_{2}$$ will form another circle $$C_{3}$$ with the same center (0,0).

## Step 4: Compute the Area Covered by $$C_{3}$$

Since $$C_{3}$$ has the same center as $$C_{1}$$ and $$C_{2}$$ and the same radius with $$C_{1}$$, the area will be $$A = \pi r^{2} = \pi*2^{2} = 4\pi$$.

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