Final answer:
To find a polynomial equation with integer coefficients for a set of coplanar points forming a parabola, we can use the distances from a point to the focus and directrix. The focus is (-3,1), and the parabola is defined as the set of points where this distance is half the distance to the line y=4. By applying the definition of a parabola to these distances and simplifying, the polynomial equation of the conic section can be derived.
Step-by-step explanation:
The task is to find a polynomial equation with integer coefficients for a set of coplanar points where the distance to the point (-3,1) is half the distance from the line y=4. This scenario describes a set of points that form a conic section, specifically a parabola.
We can start by using the definition of a parabola as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). We are given the focus (-3,1), and we know that the directrix is the line y=4. The condition that the distance from each point to the focus is half the distance to the directrix means the vertex (the halfway point between the focus and directrix) is at y=4-0.5(4-1)=3. Now we can establish the standard form of the parabola's equation.
Let's denote a point on the parabola as (x, y). The distance from (x, y) to the focus (-3, 1) is given by d1 = √[(x+3)2 + (y-1)2], and the distance to the directrix y=4 is d2 = |y-4|. According to the definition of the parabola, d1 = 1/2 * d2. Squaring both sides to eliminate the square root gives us (x+3)2 + (y-1)2 = 1/4(y-4)2. To find a polynomial equation of this conic, we need to expand and simplify this equation to get it in the form 4(x+3)2 = (y-1)2 - (y-4)2, which simplifies further to a polynomial equation.