Questions: This hyperbola is centered at the origin. Find its equation. Foci: (-2,0) and (2,0) Vertices: (-1,0) and (1,0) x^2/[?]-[]^2/[]^2=1

Transcript text: This hyperbola is centered at the origin. Find its equation. Foci: $(-2,0)$ and $(2,0)$ Vertices: $(-1,0)$ and $(1,0)$ \[ \frac{x^{2}}{[?]}-\frac{[]^{2}}{[]^{2}}=1 \]

Solution

Solution Steps

Step 1: Determine the Orientation of the Hyperbola

The hyperbola opens along the x-axis because the given orientation parameter is 'x'.

Step 2: Calculate $a$

Using the distance from the center to a vertex, $a = 1$.

Step 3: Calculate $c$

Using the distance from the center to a focus, $c = 2$.

Step 4: Solve for $b^2$

Using the relationship $c^2 = a^2 + b^2$, we find $b^2 = 3$.

Step 5: Substitute $a^2$ and $b^2$ into the General Form

The equation of the hyperbola is: \[\frac{x^2}{1} - \frac{y^2}{3} = 1\]

Final Answer:

The equation of the hyperbola, rounded to 0 decimal places where necessary, is: \[\frac{x^2}{1} - \frac{y^2}{3} = 1\]

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