Find the relation between x and y such that point (x, y) is equidistant from (3, 1) &(2, 0) 

To find the relation between \(x\) and \(y\) such that the point \((x, y)\) is equidistant from the points \((3, 1)\) and \((2, 0)\), we need to use the distance formula.

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Since the point \((x, y)\) is equidistant from both \((3, 1)\) and \((2, 0)\), the distances must be equal:

\[
\sqrt{(x - 3)^2 + (y - 1)^2} = \sqrt{(x - 2)^2 + (y - 0)^2}
\]

Now, square both sides to eliminate the square roots:

\[
(x - 3)^2 + (y - 1)^2 = (x - 2)^2 + y^2
\]

Expand both sides:

\[
(x^2 - 6x + 9) + (y^2 - 2y + 1) = (x^2 - 4x + 4) + y^2
\]

Simplify the equation by canceling out the \(x^2\) and \(y^2\) terms on both sides:

\[
-6x + 9 - 2y + 1 = -4x + 4
\]

Now simplify further:

\[
-6x - 2y + 10 = -4x + 4
\]

Move all terms involving \(x\) and \(y\) to one side:

\[
-6x + 4x - 2y = 4 - 10
\]

\[
-2x - 2y = -6
\]

Divide the equation by \(-2\) to simplify:

\[
x + y = 3
\]

Thus, the relation between \(x\) and \(y\) is:

\[
x + y = 3
\]

Would you like further details or have any questions?

Here are some related questions:
1. How can this relation be interpreted geometrically?
2. What if the points were different, say \((1, 2)\) and \((3, 4)\)?
3. How does the midpoint of \((3, 1)\) and \((2, 0)\) relate to this equation?
4. Can this problem be solved using vectors?
5. How would the solution change if we had three points instead of two?

**Tip:** When solving geometry problems, drawing a diagram can help visualize the relations and guide you through the solution.

Find the relation between x and y such that point (x, y) is equidistant from (3, 1) and (2, 0)

This problem explores how to find the relation between x and y for a point that is equidistant from two given points, (3, 1) and (2, 0), using the distance formula. Suitable for students in Grades 9-11, this example focuses on coordinate geometry and algebra.

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