### 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 foci of a hyperbola, I need to rewrite the equation in standard form.
That means completing the square for both x and y terms.
I'll start by moving the constant term to the right side: \[x^2 - 10x - 4y^2 = -5.\]
Then I'll group the x terms and the y terms together and add the appropriate values to both sides to make perfect squares: \[(x^2 - 10x + 25) - 4(y^2) = -5 + 25 - 0.\]
I can factor the x and y terms and simplify the right side: \[(x - 5)^2 - 4y^2 = 20.\]
To get the standard form, I need to divide both sides by 20: \[\frac{(x - 5)^2}{20} - \frac{y^2}{5} = 1.\]
Now I can see that the hyperbola is centered at (5, 0) and has a horizontal transverse axis.
The denominator under the x term is the square of the semi-major axis, a, and the denominator under the y term is the square of the semi-minor axis, b. So, a^2 = 20 and b^2 = 5.
The distance from the center to a focus, c, is related to a and b by the formula c^2 = a^2 + b^2.
So, c^2 = 20 + 5 = 25, and c = 5.
Since the hyperbola is horizontal, the foci are on the same horizontal line as the center, so their y-coordinates are 0.
The x-coordinates are 5 units away from the center in either direction, so they are 5 + 5 = 10 and 5 - 5 = 0.
Therefore, one of the foci is (10, 0).