If a point (x, y) in a plane is equidistant from the points (-1, 1) a…

The correct answer is Option (3) → $10x + 4y = 23$

To find the relationship between $x$ and $y$, we use the distance formula. Since the point $(x, y)$ is equidistant from $A(-1, 1)$ and $B(4, 3)$, the distance $PA$ must equal the distance $PB$.

1. Set up the Equation

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

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

Setting the distances equal ($PA = PB$):

$\sqrt{(x - (-1))^2 + (y - 1)^2} = \sqrt{(x - 4)^2 + (y - 3)^2}$

2. Simplify and Solve

Square both sides to remove the radicals:

$(x + 1)^2 + (y - 1)^2 = (x - 4)^2 + (y - 3)^2$

Expand the binomials:

$(x^2 + 2x + 1) + (y^2 - 2y + 1) = (x^2 - 8x + 16) + (y^2 - 6y + 9)$

3. Cancel Terms and Combine

Subtract $x^2$ and $y^2$ from both sides:

$2x - 2y + 2 = -8x - 6y + 25$

Move all $x$ and $y$ terms to the left side and constants to the right:

$2x + 8x - 2y + 6y = 25 - 2$

$10x + 4y = 23$

Conclusion

The correct relationship is $10x + 4y = 23$.