Question

Write an equation that states (x, y) is the same distance from (4, 1) as it is from the x-axis.

Answer 1

The equation that gives all the points (x, y) that are at the same distance from (4, 1) as from the x-axis is:

y = (1/2)*(x - 4)^2 + 1/2.

How we can find the point (x, y)?

First, remember that for a point (x, y), the distance to the x-axis is equal to y.

And we also know that the distance between two points (x₁, y₁) and (x₂, y₂) is:

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

The distance between (x, y) and (4, 1) is:

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

And this must be equal to y, so we can write:

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

This is the equation we need to solve.

We will get:

$y^2 = (x - 4)^2 + (y - 1)^2\\\\y^2 = (x - 4)^2 + y^2 - 2y + 1\\\\0 = (x - 4)^2 - 2y + 1\\\\2y = (x - 4)^2 + 1\\\\y = (1/2)*(x - 4)^2 + 1/2$

This equation gives all the points (x, y) that are at the same distance of (4, 1) as from the x-axis.

If you want to learn more about distances between points, you can read:

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