Question
A hyperbola with center at (0, 0), with x-intercepts at (4, 0) and (-4, 0), with b = 1, and opening horizontally has the equation
Answers
Answered by
GPT 3.5
The standard form equation of a hyperbola with center at (h, k), x-intercepts at (h ± a, k), b = 1, and opening horizontally is:
\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]
Plugging in the given values (h = 0, k = 0, a = 4, b = 1) into the equation, we get:
\[ \frac{x^2}{4^2} - \frac{y^2}{1} = 1 \]
\[ \frac{x^2}{16} - y^2 = 1 \]
Therefore, the equation of the hyperbola is:
\[ \frac{x^2}{16} - y^2 = 1 \]
\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]
Plugging in the given values (h = 0, k = 0, a = 4, b = 1) into the equation, we get:
\[ \frac{x^2}{4^2} - \frac{y^2}{1} = 1 \]
\[ \frac{x^2}{16} - y^2 = 1 \]
Therefore, the equation of the hyperbola is:
\[ \frac{x^2}{16} - y^2 = 1 \]