### Instruction: Find the coordinates of one of the foci of the hyperbola \[x^2 - 10x = 4y^2 - 5.\](Enter your answer as an ordered pair. Enter only one of the foci, not both.).### Response: To find the coordinates of the focus of a hyperbola, I need to rewrite the equation in standard form, which is either \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\] or \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\] depending on whether the hyperbola opens left-right or up-down, where (h,k) is the center, a is the semi-major axis, and b is the semi-minor axis.
To rewrite the equation in standard form, I need to complete the square for both x and y terms.
For the x terms, I have \[x^2 - 10x = (x-5)^2 - 25,\] so I can add 25 to both sides to get \[(x-5)^2 = 4y^2 - 5 + 25.\]
For the y terms, I have \[4y^2 = 4(y-0)^2,\] so I can divide both sides by 4 to get \[y^2 = (y-0)^2.\]
Combining these results, I get \[(x-5)^2 = 4(y-0)^2 + 20.\]
To make the right-hand side equal to 1, I need to divide both sides by 20, so I get \[\frac{(x-5)^2}{20} - \frac{(y-0)^2}{5} = 1.\]
Now I can see that the equation is in standard form, with h = 5, k = 0, a^2 = 20, and b^2 = 5.
The center of the hyperbola is (5,0), and the distance from the center to the focus is c, where c^2 = a^2 + b^2.
Plugging in the values of a^2 and b^2, I get \[c^2 = 20 + 5 = 25,\] so \[c = \sqrt{25} = 5.\]
Since the hyperbola opens left-right, the focus is either to the left or right of the center by c units.
Therefore, the coordinates of the focus are either (5 + c, 0) or (5 - c, 0).
Plugging in c = 5, I get (10, 0) or (0, 0) as the possible foci.
I can enter either one as the answer.