Find the standard form of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse axis: (0, −6), (0, 6); asymptote: y=2x

Verified step by step guidance

1. Identify the center of the hyperbola by finding the midpoint of the transverse axis endpoints. The midpoint formula is \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Using the points \((0, -6)\) and \((0, 6)\), calculate the center.
2. Determine the length of the transverse axis, which is the distance between the endpoints. Use the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), then find \(a\), which is half of this length.
3. Since the transverse axis is vertical (both endpoints have the same x-coordinate), the standard form of the hyperbola is \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\), where \((h, k)\) is the center.
4. Use the given asymptote equation \(y = 2x\) to find the relationship between \(a\) and \(b\). For a vertical transverse axis hyperbola, the asymptotes have equations \(y - k = \pm \frac{a}{b}(x - h)\). Match the slope \(\pm \frac{a}{b}\) to the slope of the given asymptote to solve for \(b\).
5. Substitute the values of \(a\), \(b\), and the center \((h, k)\) into the standard form equation \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\) to write the equation of the hyperbola in standard form.

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola's equation depends on the orientation of its transverse axis. For a vertical transverse axis centered at (h, k), the equation is ((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1. Understanding this form helps in writing the equation once the center, a, and b are known.

Transverse Axis and Its Endpoints

The transverse axis of a hyperbola is the line segment that passes through the two vertices. The endpoints given, (0, -6) and (0, 6), indicate the vertices and help determine the center (midpoint) and the distance a (half the length of the transverse axis), which are essential for forming the equation.

Asymptotes of a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches. Their slopes relate to the values of a and b in the hyperbola's equation. For a vertical transverse axis, the asymptotes have equations y - k = ±(a/b)(x - h). Using the given asymptote y = 2x helps find the ratio a/b and thus b.