Three distinct point A, B and C are given in the 2-dimensional coordinates 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

Options

A. (a) `((5)/(4),0)`
B. (b) `((5)/(2),0)`
C. (c) `((5)/(3),0)`
D. (d) `(0, 0)`

Answer

To solve the problem, we need to find the circumcenter of triangle ABC given the ratio of distances from a point to two fixed points. Let's break down the solution step by step.

### Step 1: Understand the Given Information
We have three distinct points A, B, and C in the 2D coordinate plane. The ratio of the distance from any one of these points (let's say point P) to the point (1, 0) and the distance from point P to the point (-1, 0) is given as \( \frac{1}{3} \).

### Step 2: Set Up the Distance Ratio
Let the coordinates of point P be \( (h, k) \). The distance from point P to (1, 0) is given by:
\[ d_1 = \sqrt{(h - 1)^2 + (k - 0)^2} = \sqrt{(h - 1)^2 + k^2} \]
The distance from point P to (-1, 0) is:
\[ d_2 = \sqrt{(h + 1)^2 + (k - 0)^2} = \sqrt{(h + 1)^2 + k^2} \]
According to the problem, the ratio of these distances is:
\[ \frac{d_1}{d_2} = \frac{1}{3} \]

### Step 3: Square Both Sides
Squaring both sides of the equation gives:
\[ \frac{(h - 1)^2 + k^2}{(h + 1)^2 + k^2} = \frac{1}{9} \]
Cross-multiplying yields:
\[ 9((h - 1)^2 + k^2) = (h + 1)^2 + k^2 \]

### Step 4: Expand and Simplify
Expanding both sides:
\[ 9(h^2 - 2h + 1 + k^2) = h^2 + 2h + 1 + k^2 \]
This simplifies to:
\[ 9h^2 - 18h + 9 + 9k^2 = h^2 + 2h + 1 + k^2 \]
Rearranging terms gives:
\[ 8h^2 - 20h + 8k^2 + 8 = 0 \]

### Step 5: Divide by 8
Dividing the entire equation by 8 simplifies it:
\[ h^2 - \frac{5}{2}h + k^2 + 1 = 0 \]

### Step 6: Complete the Square
To complete the square for the \( h \) terms:
\[ h^2 - \frac{5}{2}h = \left(h - \frac{5}{4}\right)^2 - \frac{25}{16} \]
Substituting back gives:
\[ \left(h - \frac{5}{4}\right)^2 + k^2 - \frac{25}{16} + 1 = 0 \]
Converting 1 to a fraction:
\[ \left(h - \frac{5}{4}\right)^2 + k^2 - \frac{25}{16} + \frac{16}{16} = 0 \]
This simplifies to:
\[ \left(h - \frac{5}{4}\right)^2 + k^2 = \frac{9}{16} \]

### Step 7: Identify the Circle
The equation \( \left(h - \frac{5}{4}\right)^2 + k^2 = \frac{9}{16} \) represents a circle centered at \( \left(\frac{5}{4}, 0\right) \) with a radius of \( \frac{3}{4} \).

### Step 8: Conclusion
Since points A, B, and C lie on this circle, the circumcenter of triangle ABC is the center of this circle.

Thus, the circumcenter of triangle ABC is at the point:
\[ \boxed{\left(\frac{5}{4}, 0\right)} \]