### Instruction: A hyperbola centered at the origin has foci at $(\pm 7, 0),$ and passes through the point $(2, 12).$ If the equation of the hyperbola is \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]where $a$ and $b$ are positive constants, compute the ordered pair $(a, b).$.### Response: I recall that for any hyperbola, the distance between the foci is $2c,$ where $c$ is related to $a$ and $b$ by the equation $c^2 = a^2 + b^2.$
In this case, $2c = 14,$ so $c = 7,$ and $c^2 = 49.$
I also recall that for any point $(x, y)$ on the hyperbola, the difference between the distances to the foci is a constant, equal to $2a.$
In this case, the point $(2, 12)$ satisfies this condition, so I can use the distance formula to find $2a.$
The distance from $(2, 12)$ to $(7, 0)$ is $\sqrt{(2 - 7)^2 + (12 - 0)^2} = \sqrt{169} = 13.$
The distance from $(2, 12)$ to $(-7, 0)$ is $\sqrt{(2 + 7)^2 + (12 - 0)^2} = \sqrt{225} = 15.$
Therefore, the difference between the distances is $15 - 13 = 2,$ and this must be equal to $2a.$
Solving for $a,$ I get $a = 1.$
Now I can plug in $a$ and $c$ into the equation $c^2 = a^2 + b^2$ and solve for $b.$
I get $49 = 1 + b^2,$ so $b^2 = 48,$ and $b = \sqrt{48}.$
Since $b$ must be positive, I can ignore the negative root.
Therefore, the ordered pair $(a, b)$ is $(1, \sqrt{48}).$