Mathematics worksheet for identifying conic sections such as parabolas, circles, ellipses, and hyperbolas through equations and graphs.

To identify the conic sections in the given problem, we need to analyze each equation and graph provided. Conic sections include parabolas, circles, ellipses, and hyperbolas. Let's go through each one step by step.

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Step 1: Analyze the Equations

#### a- \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \)
- This is an ellipse because it is in the standard form of an ellipse:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Here, \( a^2 = 4 \) and \( b^2 = 16 \). Since \( b^2 > a^2 \), the major axis is vertical.

#### b- \( \frac{x^2}{1} + \frac{y^2}{36} = 1 \)
- This is also an ellipse because it is in the standard form of an ellipse:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Here, \( a^2 = 1 \) and \( b^2 = 36 \). Since \( b^2 > a^2 \), the major axis is vertical.

#### c- \( x^2 + y^2 = 16 \)
- This is a circle because it is in the standard form of a circle:
\[
x^2 + y^2 = r^2
\]
Here, \( r^2 = 16 \), so \( r = 4 \).

#### d- \( y = x^2 \)
- This is a parabola because it is in the standard form of a parabola that opens upwards:
\[
y = ax^2 + bx + c
\]
Here, \( a = 1 \), \( b = 0 \), and \( c = 0 \).

#### e- (Graph of an ellipse)
- The graph shows a closed curve that is symmetric about both axes and has two foci. This is an ellipse.

#### f- (Graph of a hyperbola)
- The graph shows two separate branches that are symmetric about both axes. This is a hyperbola.

#### g- \( \frac{(x-3)^2}{121} + \frac{(y+2)^2}{4} = 1 \)
- This is an ellipse because it is in the standard form of an ellipse:
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]
Here, \( h = 3 \), \( k = -2 \), \( a^2 = 121 \), and \( b^2 = 4 \). Since \( a^2 > b^2 \), the major axis is horizontal.

#### h- \( \frac{y^2}{9} - \frac{x^2}{16} = 1 \)
- This is a hyperbola because it is in the standard form of a hyperbola:
\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\]
Here, \( a^2 = 9 \) and \( b^2 = 16 \). The transverse axis is vertical.

#### i- (Graph of a hyperbola)
- The graph shows two separate branches that are symmetric about both axes. This is a hyperbola.

#### j- \( y = x^2 \)
- This is a parabola because it is in the standard form of a parabola that opens upwards:
\[
y = ax^2 + bx + c
\]
Here, \( a = 1 \), \( b = 0 \), and \( c = 0 \).

#### k- \( y = a(x-h)^2 + k \)
- This is a parabola because it is in the vertex form of a parabola:
\[
y = a(x-h)^2 + k
\]
The value of \( a \) determines the direction and width of the parabola.

#### l- \( (x+3)^2 + (y-4)^2 = 12 \)
- This is a circle because it is in the standard form of a circle:
\[
(x-h)^2 + (y-k)^2 = r^2
\]
Here, \( h = -3 \), \( k = 4 \), and \( r^2 = 12 \), so \( r = \sqrt{12} \).

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Step 2: Summarize the Results

- a- Ellipse
- b- Ellipse
- c- Circle
- d- Parabola
- e- Ellipse
- f- Hyperbola
- g- Ellipse
- h- Hyperbola
- i- Hyperbola
- j- Parabola
- k- Parabola
- l- Circle

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Final Answer

\[
\boxed{
\begin{array}{ccc}
\text{a-} & \text{Ellipse} & \\
\text{b-} & \text{Ellipse} & \\
\text{c-} & \text{Circle} & \\
\text{d-} & \text{Parabola} & \\
\text{e-} & \text{Ellipse} & \\
\text{f-} & \text{Hyperbola} & \\
\text{g-} & \text{Ellipse} & \\
\text{h-} & \text{Hyperbola} & \\
\text{i-} & \text{Hyperbola} & \\
\text{j-} & \text{Parabola} & \\
\text{k-} & \text{Parabola} & \\
\text{l-} & \text{Circle} & \\
\end{array}
}
\]