Multiple choice

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 of the curves ${C}{2}$ is the curve ${C}{3}$. Then

  1. ${C}_{3}$ is a circle
  2. The area enclosed by the curve ${C}_{3}$ is $8\pi$
  3. ${C}_{2}$ and ${C}_{3}$ are circles with the same centre
  4. Radian of ${C}_{3}$ is $8$
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AI explanation

The curve C2 is the locus of the intersection of orthogonal tangents to the circle C1 (x^2 + y^2 = 4), which is defined as its director circle with the equation x^2 + y^2 = 2 * 2^2, or x^2 + y^2 = 8. The curve C3 is the director circle of C2, found by taking twice its radius squared, resulting in the equation x^2 + y^2 = 16. Because the equation of C3 represents a constant distance from the origin, the curve C3 is a circle.

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