Question

Write an equation representing the set of all points that is equidistant from the line y=1 and the point (0,3)

Answer 1

This is a paraola: y = 1 is the directrix (it is perpendicular to the axis of symmetry); the focus is (0, 3)

The vertex of the parabola (which must also be on the parabola) is at (0, 1) because it is at a distance of 1 away; it is the nearest point to the point y = 1 & is at a distance 1 away from (0, 3)

The distance of the point (x,y) from the point (0,3) is gotten by using the formula for distance between two points (Pythagoras theorem):

`\begin{gathered} \sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}\Rightarrow\sqrt[]{(x-0)^2+(y-3)^2} \\ \text{the distance from y=1 is given by y-1} \\ (y-1)^2=x^2+(y-3)^2 \\ x^2=(y-1)^2-(y-3)^2 \\ x^2=\lbrack y(y-1)-1(y-1)\rbrack-\lbrack y(y-3)-3(y-3)\rbrack \\ x^2=y^2-y-y+1-(y^2-3y-3y+9) \\ x^2=y^2-y^2-2y-(-6y)+1-9 \\ x^2=4y-8 \\ 4y=x^2+8\Rightarrow y=(1)/(4)x^2+2 \\ y=(1)/(4)x^2+2 \end{gathered}`
`\begin{gathered} y=0.25x^2+2 \\ x=-2 \\ y=0.25(-2^2)+2=0.25\cdot4+2=3 \\ x=-1,y=2.25 \\ x=0,y=2 \\ x=1,y=2.25 \\ x=2,y=3 \end{gathered}`