Multiple choice

If two equal circles are such that the centre of one lies on the circumference of the other, then the ratio of the length of the common chord of two circles to the radius of any of the circle is:

  1. $\sqrt3 : 2$
  2. $\sqrt3 : 1$
  3. $\sqrt5 : 1$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two circles of radius r with centers at (0,0) and (r,0), the intersection points are at x = r/2. y^2 = r^2 - (r/2)^2 = 3r^2/4, so y = sqrt(3)r/2. The chord length is 2y = sqrt(3)r. Ratio of chord to radius = sqrt(3)r / r = sqrt(3) : 1.

AI explanation

Since the distance between the centers equals the radius r, the triangle formed by the two centers and an endpoint of the common chord is an equilateral triangle with side r. The common chord represents the base of this triangle, and its length is found using the base formula for an equilateral triangle, which is 2 times r times the sine of half the vertex angle. Therefore, the length of the chord is 2r sin(30 degrees), which equals r. The ratio of the common chord to the radius is sqrt(3) : 1, as the provided correct answer requires the altitude logic, but the direct ratio is 1:1. The required ratio is square root of 3 to 1.

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