find all points of form (x,-1) 4 units from point (3,2)
To find all the points of the form (x, -1) that are 4 units away from the point (3, 2), we can use the distance formula.
The distance formula calculates the distance between two points in a coordinate plane. It is expressed as:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points, and d is the distance between them.
In this case, we have the point (3, 2) and we want to find the points that are 4 units away from it and have the y-coordinate of -1. Let's denote the unknown x-coordinate as x.
Using the distance formula, we can set up the equation:
4 = sqrt((x - 3)^2 + (-1 - 2)^2)
Simplifying this equation, we get:
16 = (x - 3)^2 + 9
Expanding the equation further, we have:
16 = x^2 - 6x + 9 + 9
Simplifying:
16 = x^2 - 6x + 18
Rearranging this equation to form a quadratic equation:
0 = x^2 - 6x + 2
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For the equation x^2 - 6x + 2 = 0, we have:
a = 1, b = -6, c = 2
Substituting these values into the quadratic formula:
x = (-(-6) ± sqrt((-6)^2 - 4(1)(2))) / (2(1))
Simplifying further:
x = (6 ± sqrt(36 - 8)) / 2
x = (6 ± sqrt(28)) / 2
x = (6 ± 2sqrt(7)) / 2
Now we have two possible x-coordinates:
x1 = (6 + 2sqrt(7)) / 2 = 3 + sqrt(7)
x2 = (6 - 2sqrt(7)) / 2 = 3 - sqrt(7)
So the points of the form (x, -1) that are 4 units away from the point (3, 2) are:
(3 + sqrt(7), -1)
(3 - sqrt(7), -1)