Write the equation of the hyperbola $16 x^{2}-4 y^{2}-32 x-16 y-64=0$ in standard form Where: \[ \frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1, \] \[ \begin{array}{l} h= \\ k= \\ a= \\ b= \end{array} \] Question Help: Video Message instructor Submit Question
Solution
Step 1 ：Given the equation of the hyperbola \(16 x^{2}-4 y^{2}-32 x-16 y-64=0\), we want to rewrite it in standard form \(\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1\), and find the values of h, k, a, and b.
Step 2 ：First, we complete the square for the equation, which gives us \(x^{2}/4 - x/2 - y^{2}/16 - y/4 - 1\).
Step 3 ：We then simplify the equation by dividing all terms by 64, which gives us \(x^{2}/4 - x/2 - y^{2}/16 - y/4 - 1\).
Step 4 ：From this equation, we can extract the values of h, k, a, and b. We find that \(h = 1/4\), \(k = 1/8\), \(a = 2\), and \(b = 4\).
Step 5 ：Final Answer: The equation of the hyperbola in standard form is \(\frac{(x-1)^{2}}{4}-\frac{(y-1/2)^{2}}{16}=1\). The values of h, k, a, and b are \(h=\boxed{1}\), \(k=\boxed{\frac{1}{2}}\), \(a=\boxed{2}\), and \(b=\boxed{4}\).
