### 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 write it in standard form, either \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) or \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\), where \(h\) and \(k\) are the coordinates of the center, \(a\) and \(b\) are the semi-major and semi-minor axes, and \(c\) is the distance from the center to the foci.
To write the given equation in standard form, I need to complete the square for both \(x\) and \(y\) terms.
For the \(x\) terms, I have \(x^2 - 10x\), which has a coefficient of 1 for \(x^2\).
To complete the square, I need to add and subtract \(\frac{(-10)^2}{4} = 25\) inside the parentheses.
So I get \((x^2 - 10x + 25) - 25\), which is \((x-5)^2 - 25\).
For the \(y\) terms, I have \(4y^2 - 5\), which has a coefficient of 4 for \(y^2\).
To complete the square, I need to factor out 4 and add and subtract \(\frac{(-0)^2}{4} = 0\) inside the parentheses.
So I get \(4(y^2 - \frac{5}{4}) + 0\), which is \(4(y-0)^2 - 5\).
Now I can rewrite the equation as \((x-5)^2 - 25 = 4(y-0)^2 - 5\), and move the constant terms to the right side.
I get \((x-5)^2 - 4(y-0)^2 = 20\).
To make the right side equal to 1, I need to divide both sides by 20.
I get \(\frac{(x-5)^2}{20} - \frac{4(y-0)^2}{20} = 1\).
This is almost in standard form, but I need to simplify the fractions.
I get \(\frac{(x-5)^2}{20} - \frac{(y-0)^2}{5} = 1\).
Now I can see that the center of the hyperbola is \((5,0)\), the semi-major axis is \(\sqrt{20}\), the semi-minor axis is \(\sqrt{5}\), and the hyperbola opens horizontally.
To find the distance \(c\) from the center to the foci, I need to use the relationship \(c^2 = a^2 + b^2\) for a horizontal hyperbola.
I get \(c^2 = 20 + 5\), which is \(c^2 = 25\).
Taking the positive square root, I get \(c = 5\).
Since the center is \((5,0)\) and the hyperbola opens horizontally, the foci are \((5 \pm c, 0)\).
Plugging in \(c = 5\), I get \((5 \pm 5, 0)\), which are \((10,0)\) and \((0,0)\).
The question asks for only one of the foci, so I can choose either one.
I'll pick \((10,0)\).