The distance between two points in the coordinate plane can be calculated using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points.In this case, we have:
d = 19
√((x2 - 3)^2 + (y2 + 3)^2) = 19
Squaring both sides, we get:
(x2 - 3)^2 + (y2 + 3)^2 = 361
Expanding, we get:
x^2 - 6x + 9 + y^2 + 6y + 9 = 361
x^2 - 6x + y^2 + 6y = 343
This is the equation of a circle with center (3, -3) and radius 19.The set of all such that the distance between the points (3, -3) and (−6,) is 19 is the set of all points on this circle.To find the equation in standard form, we can complete the square.Moving the constant term to the right side of the equation, we get:
x^2 - 6x + y^2 + 6y = 343
x^2 - 6x + y^2 + 6y - 343 = 0
We can complete the square in x by adding (−6/2)2=9 to both sides of the equation.
x^2 - 6x + 9 + y^2 + 6y - 343 = 9
x^2 - 6x + y^2 + 6y - 352 = 0
We can complete the square in y by adding (6/2)2=9 to both sides of the equation.
(x - 3)^2 + (y + 3)^2 = 343