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So, I'm given a line , $y=-1$ , and a point $(1,0)$ , and I have to find the points that have the same distance between these two.

Now I know that the distance formula requires two points, so the question is how I can find a point on the line $y=-1$? So I can find the set of points?

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    Hint: look up the geometric definition of a parabola. Wikipedia will help.: https://en.wikipedia.org/wiki/Parabola – 2017-01-28

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We know that the distance of a point $(x_0, y_0) $ from a line $ax+by+c=0$ is given by $\frac {|ax_0+ by_0 +c|}{\sqrt {a^2+b^2}} $.

Using this formula we get $$ y_0 +1 = \sqrt {(x_0-1)^2 +y_0^2} $$ $$\Rightarrow x_0^2 =2 (x_0 +y_0) $$ is the required locus of the points satisfying the condition. Hope it helps.