Multiple choice

The length of the common chord of the ellipse $\cfrac { { (x-1) }^{ 2 } }{ 9 } +\cfrac { { (y-2) }^{ 2 } }{ 4 } =1$ and the circle $ { (x-1) }^{ 2 }+{ (y-2) }^{ 2 }=1$ is:

  1. $1$
  2. $3$
  3. $8$

  4. None of these
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D Correct answer
Explanation

The ellipse is ((x-1)^2)/9 + ((y-2)^2)/4 = 1 and the circle is (x-1)^2 + (y-2)^2 = 1. Since the circle is entirely inside the ellipse (except at points where the ellipse is smaller, but here the ellipse is larger in both axes), they do not intersect. The common chord length is undefined or 0.

AI explanation

Both the ellipse and the circle are perfectly centred at (1, 2). Because they share the same centre, their common chord must be a diameter, and any diameter of the circle has a length of 2 times its radius. Since the circle has a radius of 1, the length of the common chord is 2. Because this value is not listed among the choices, the correct option is None of these.

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