equations from geometrical defintions?

Question

find a polynomial equation with integer coefficients for the set of coplanar points described.

Problem: each point is equidistant from the point (2,5) and the line x= -3.

my answer is y^2-10x-10y+20=0

is this right?

Answer 1

It is a parabola with directrix x = -3, focus = (2,5) and vertex at (-.5,5).
so its equation is (y- 5)^2 = 2p (x-(-1/2))
(y-5)^2 =5(x +1/2)
If you expand this you get: y^2-10y+25 =5x+5/2

Answer 2

Sounds like a parabola to me. Do you have the general equation for a parabola?

x = 1/(4p) ( y - k)^2 + h,

Where (h,k) is the vertex of the parabola (the point that is dead in between the line (called the directrix) and the point (called the focus))

and p is the directed distance (ie positive or negative in the direction you're counting from) from the directrix to the vertex.

If you know your general forms for a parabola, then all of these algebra problems are reduced to counting problems on a piece of graph paper.

Good luck!

Answer 3

This is a parabola with directrix x = -3, focus = (2,5) and vertex at (-.5,5).
Thus its equation is (y- 5)^2 = 2p (x-(-.5))
(y-5)^2 =5(x +.5) [2p= distance between focus and directrix.]
If you expand this you get y^2-10y+25 =5x+2.5
y^2 -10y -5x +22.5 = 0