In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $16x^2-y^2+64x-2y+67=0$

Verified step by step guidance

1. Start with the given equation: \$16x^2 - y^2 + 64x - 2y + 67 = 0$.
2. Group the \(x\) terms and \(y\) terms together: \((16x^2 + 64x) - (y^2 + 2y) = -67\).
3. Factor out the coefficient of \(x^2\) from the \(x\) terms: \$16(x^2 + 4x) - (y^2 + 2y) = -67$.
4. Complete the square for both \(x\) and \(y\) terms inside the parentheses: - For \(x^2 + 4x\), take half of 4 (which is 2), square it (4), and add and subtract inside the parentheses. - For \(y^2 + 2y\), take half of 2 (which is 1), square it (1), and add and subtract inside the parentheses.
5. Rewrite the equation including the completed squares and adjust the constant term accordingly, then divide through by the constant to get the equation in standard form of a hyperbola.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x + p)^2 or (y + q)^2 by adding and subtracting appropriate constants. This technique helps convert the given equation into a standard form, making it easier to identify the conic section and its properties.

Standard Form of a Hyperbola

The standard form of a hyperbola is an equation that clearly shows its center, orientation, and shape, typically written as (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or vice versa. Converting to this form allows for straightforward identification of key features like vertices, foci, and asymptotes.

Foci and Asymptotes of a Hyperbola

The foci are two fixed points that define the hyperbola, located along the transverse axis, and are essential for its geometric definition. Asymptotes are lines that the hyperbola approaches but never touches, given by linear equations derived from the hyperbola's standard form, guiding the graph's shape.