jmc

Let $F_1=(7,0)$ and $F_2=(-7,0)$ be the two foci. We know that if the hyperbola has equation $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$$then for any point $P$ on the hyperbola, we have $|P{F}_1-P{F}_2|=2a.$ We are given that the point $P=(2,12)$ lies on the hyperbola. We have $P{F}_1=\sqrt{(7-2{)}^2+(12-0{)}^2}=13$ and $P{F}_2=\sqrt{(-7-2{)}^2+(12-0{)}^2}=15,$ so $|P{F}_1-P{F}_2|=|13-15|=2.$ Thus, $2=2a,$ so $a=1.$

Now, the distance from the center of the hyperbola (the origin) to each focus is $7,$ so we have $a^2+b^2=7^2=49.$ Substituting $a=1,$ we get $1+b^2=49,$ so $b=\sqrt{48}=4\sqrt{3}.$ Thus, $(a,b)=\boxed{(1,4\sqrt{3})}.$