Multiple choice

Three distinct points A, B and C are given in the 2- dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (-1, 0) is equal to 1/3. Then the circumcentre of the triangle ABC is at the point

  1. $(\frac { 5 }{ 4 } ,0)$
  2. $(\frac { 5 }{ 2 } ,0)$
  3. $(\frac { 5 }{ 3 } ,0)$

  4. (0, 0)
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A Correct answer
Explanation

The locus of points whose distances from (1, 0) and (-1, 0) are in ratio 1/3 is a circle (Apollonius circle). The circumcenter of any triangle formed by points on this circle is the center of this circle, which lies on the x-axis at x = 5/4.

AI explanation

Let any point be (x, y) with the given distance ratio of 1/3 to (1, 0) and (-1, 0). Using the distance formula, the square root of (x - 1) squared + y squared divided by the square root of (x + 1) squared + y squared equals 1/3. Squaring and cross-multiplying gives 9((x - 1) squared + y squared) = (x + 1) squared + y squared, which simplifies to 8x squared + 8y squared - 20x + 8 = 0, or x squared + y squared - 2.5x + 1 = 0. This equation represents a circle, and its center is (1.25, 0) or (5/4, 0), which is the circumcentre of triangle ABC.

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