Problem. Find the domain of the parabola with focus $(1, 2)$ and directrix $2x+y=1$.
My attempt
Using the distance from a point to a line formula and the point-to-point distance formula, I have gotten this equation:
$$\frac{\left|2x+y-1\right|}{\sqrt{5}}=\sqrt{\left(x-1\right)^{2}+\left(y-2\right)^{2}}.$$
Clearly, simplifying this would be a pain and I'm not sure it would help even if I did.
By graphing the parabola on desmos, we find that domain is $\{x|x\ge\frac14\}$. When I try to plug in smaller values such as $\frac15$, I get
$$\frac{|\frac25+y-1|}{\sqrt5}=\sqrt{\left(\frac15-1\right)^2+(y-2)^2}.$$
I am pretty sure simplifying and solving would yield no real solutions for $y$, but I want
- An elegant solution(that does not involve a ton of bashing algebra)
- A method of finding the domain that does not rely on graphing.
Thanks!
Best Answer
The simplest method I see, is actually to simplify $$\frac{\left|2x+y-1\right|}{\sqrt{5}}=\sqrt{\left(x-1\right)^{2}+\left(y-2\right)^{2}}$$
$$\implies5x^2-10x+5y^2-20y+25=4x^2+y^2+1+4xy-4x-2y$$
Rewrite, treating it as a quadratic in $y$
$$4y^2+y(-4x-18)+(x^2-6x+4)=0$$
We wish a value of $x$, for which we have at least one unique solution to $y$ $$\implies \Delta\ge0$$ $$\implies(-4x-18)^2-4\cdot(4)\cdot(x^2-6x+24)\ge0$$ $$\implies4x^2+36x+81-4x^2+24x-96\ge0\implies \boxed{x\ge\frac14}$$