I need to determine the equation of a circle that:
Touches the x axis
Has a point R=[2;1]
I know I need to get the location of the middle and size of the radius to create the equation, but I am at loss of finding a way to do that.
I would probably be able to do it myself if I had some more information about the circle (like radius) but I've already spent a lot of time trying to figure out how to solve it with so little information. Any help is appreciated.
As many people have commented, there’s not enough information here to find a unique solution. If this is really the problem you’ve been given in its entirety, the best that you can do is find a family of circles that satisfy the constraints.
Since the circle is tangent to the $x$-axis, its radius must be equal to the absolute value of the center’s $y$-coordinate. Thus, the equation of a circle that fits these constraints will have the form $$(x-h)^2+(y-k)^2=k^2.$$ Plugging in the known point will give you a constraint on $h$ and $k$, the coordinates of the center: $$(2-h)^2+(1-k)^2 = k^2 \\ h^2+k^2-2h-2k+5 = k^2 \\ k = \frac12(h^2-4h+5)$$ therefore, any circle with a center on the parabola $\frac12(x^2-4x+5)$ with radius equal to the $y$-coordinate satisfies the problem’s constraints.