Question

.Let F = (3,2). There is a point P on the y-axis for which the distance from P to the x-axis equals the distance PF. Find the coordinates of P.

Answer 1

The coordinates of point P are
`\((0, (13)/(4))\).`

Let's denote the coordinates of point P on the y-axis as (0, y). The distance from P to the x-axis is then given by the y-coordinate (since P lies on the y-axis), and the distance from P to point F (3, 2) is given by the distance formula:

`\[ PF = √((x_2 - x_1)^2 + (y_2 - y_1)^2) \]`

In this case,
`\( x_1 = 0, y_1 = y, x_2 = 3, \)`and
`\( y_2 = 2 \).` Therefore, the distance formula becomes:

`\[ PF = √((3 - 0)^2 + (2 - y)^2) \]`

We want the distance from P to the x-axis to equal PF. The distance from P to the x-axis is simply the y-coordinate of P, which is y. So, we have the equation:

`\[ y = √((3 - 0)^2 + (2 - y)^2) \]`

Solving for y:

`\[ y = √(9 + (2 - y)^2) \]`

`\[ y^2 = 9 + (2 - y)^2 \]`

`\[ y^2 = 9 + 4 - 4y + y^2 \]`

`\[ 4y = 13 \]`

`\[ y = (13)/(4) \]`

So, the coordinates of point P are
`\((0, (13)/(4))\).`