To determine the true statements about the given conic section:
[tex]\[
\frac{(y+4)^2}{81}-\frac{(x+3)^2}{9}=1
\][/tex]
Let's analyze the given equation and identify its key features:
1. Identify the type of conic section: The equation provided is in the form
[tex]\[
\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
\][/tex]
With a positive term and a negative term, the equation represents a hyperbola, not an ellipse.
2. Determine the center: In the standard form of the hyperbola
[tex]\[
\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
\][/tex]
the center is given by the coordinates [tex]\((h, k)\)[/tex]. In the given equation, [tex]\((h, k)\)[/tex] can be identified by comparing it to the formula:
- Here, [tex]\((x+3)\)[/tex] can be rewritten as [tex]\((x - (-3))\)[/tex], so [tex]\(h = -3\)[/tex].
- Similarly, [tex]\((y+4)\)[/tex] can be rewritten as [tex]\((y - (-4))\)[/tex], so [tex]\(k = -4\)[/tex].
Therefore, the center is [tex]\((-3, -4)\)[/tex].
3. Determine the direction in which the hyperbola opens: Since the positive term is associated with the [tex]\((y+4)^2\)[/tex] term, this indicates that the hyperbola opens vertically (upward and downward), rather than horizontally (left and right).
Based on these analyses:
- The center of the hyperbola is [tex]\((-3, -4)\)[/tex], making statement b true.
- The hyperbola opens vertically, so statement c which says it opens to the left and right is false.
- The graph is a hyperbola, making statement d false, which suggests it is an ellipse.
The true statements are:
b. Center [tex]$(-3,-4)$[/tex]
