Every nondegenerate quadratic form defined on a two dimensional space over a finite field is universal.

I want to prove the following theorem:

Let $F$ be a field.A quadratic form $q:V\to F$ is called universal if it takes all values in $F^\times$.Show that every nondegenerate quadratic form defined on a two dimensional space over a finite field is universal.

I am new in quadratic forms and I have no idea how to solve it.Can someone show me a proof?

We let $F=\mathbb{F}_q$ and consider the quadratic form $(x,y)\to ax^2+by^2$ where $a,b$ is not zero in $\mathbb{F}_q$. We show $ax^2+by^2=c$ has a solution. If not, then $\{ax^2\}$ and $\{c-by^2\}$ has no intersection in $\mathbb{F}_q$. But we know there are ${q+1\over 2}$ squares in $\mathbb{F}_q$, so they cannot have void intersection.