Two subsets of the plane?

Question

Aren't planes described by $(x,y,z)$?

I know that the point (2,0) satisfies a, and the point (1.5,4) satisfies b. But how would I turn these into equations and geometric descriptions?

I'm stuck.

Answer 1

By the Pythagorean theorem, the distance between two points $(x,y)$ and $(a,b)$ is given by $\sqrt{(x-a)^2+(x-b)^2}$. The condition given in a) then translates to $$\sqrt{(x-4)^2+y^2}=\sqrt{(x-1)^2+y^2}.$$ We can solve this for $y$. Take the square and simplify to obtain $$y=\pm\sqrt{4-x^2}.$ This equation characterizes $S$. It is a circle around the origin with radius $2$.

Answer 2

Hint: When trying to form equations from descriptions, "is" usually corresponds to an equal sign. "The distance between a point $(a,b)$ and a point $(c,d)$" is the quantity $\sqrt{(a-c)^2+(b-d)^2}$. Do you know how to find the distance between a point and a vertical line?

Answer 3

Hint: Using the distance formula, a point $(x,y)$ will satisfy a) exactly when $$ \sqrt{(x-4)^2 + y^2} = 2\sqrt{(x-1)^2 + y^2} = 1 $$ and it will satisfy b) if $$ |x+1| = \sqrt{(x-2)^2 + (y-4)^2} $$ However, both of these equations can be simplified.