Find a function to represent the set of all points equidistant from the point(-2,3) and the line y=7.
That line is a parabola. (-2,3) is the focus. y=7 is called the directrix. You probably have some text or lecture notes that tell you how to relate those facts to the equation you need. We will be glad to critique your work.
What's function to represent the set of all points equidistant from the point
(-2,3) and the line y=7?
I have suggested you review how to relate the equation of a parabola to the position of the focus and the directrix. You must have a section in your text that discusses this. If there is none, I suggest you review
http://home.alltel.net/okrebs/page64.html
your mother
To find a function that represents the set of all points equidistant from the point (-2,3) and the line y=7, we can use the distance formula.
The distance between a point (x,y) and a line Ax + By + C = 0 is given by:
d = |Ax + By + C| / √(A² + B²)
In our case, the line equation is y = 7, which can be rewritten as 0x + 1y - 7 = 0.
Using the distance formula, we have:
d = |0x + 1y - 7| / √(0² + 1²)
d = |y - 7| / √1
d = |y - 7|
Now, we want to find the set of all points equidistant from (-2,3) and y=7. Let (x,y) be a point in this set. The distance between (-2,3) and (x,y) should be equal to the distance between (x,y) and y=7. Therefore, we get the equation:
√[(x - (-2))² + (y - 3)²] = |y - 7|
Simplifying this equation, we have:
(x + 2)² + (y - 3)² - (y - 7)² = 0
Expanding and combining like terms, we have:
x² + 4x + 4 + y² - 6y + 9 - y² + 14y - 49 = 0
Simplifying further, we get:
x² + 4x - 16y - 36 = 0
Therefore, the function that represents the set of all points equidistant from the point (-2,3) and the line y=7 is:
f(x,y) = x² + 4x - 16y - 36