Question

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

Accepted Answer

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. 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. 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. 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 $$b. $$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.

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 video answer for a similar problem:

Key Concepts

Standard Form of a Hyperbola

Transverse Axis and Its Endpoints

Asymptotes of a Hyperbola