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Drag the tiles to the boxes to form correct pairs. Not all tiles will be used.

Match the equations of ellipses to their corresponding pairs of vertices.

[tex]\[
\begin{array}{l}
\frac{(x-3)^2}{3^2}+\frac{(y+4)^2}{2^2}=1 \quad \text{(0,-4) and (6,-4)} \\
\frac{(x+7)^2}{7^2}+\frac{(y+2)^2}{4^2}=1 \quad \text{(-3,-6) and (11,-6)} \\
\frac{(x-1)^2}{2^2}+\frac{(y-7)^2}{6^2}=1 \\
\frac{(x-4)^2}{7^2}+\frac{(y+6)^2}{5^2}=1 \quad \text{(-5,-4) and (-5,12)} \\
\frac{(x+5)^2}{5^2}+\frac{(y-4)^2}{8^2}=1 \quad \text{(-1,-10) and (-1,12)} \\
\frac{(x+1)^2}{9^2}+\frac{(y-1)^2}{11^2}=1 \\
\end{array}
\][/tex]

[tex]\[
\square \longleftrightarrow \\
\][/tex]

Sagot :

GinnyAnswer
To determine the pairs of vertices for each ellipse equation, we need to recognize the standard form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h, k)\) represents the center of the ellipse, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. We then identify the vertices, which are located at \((h \pm a, k)\) and \((h, k \pm b)\) depending on whether the major axis is horizontal or vertical.

Let's analyze each equation and match them with their respective pairs of vertices.

1. Equation: \(\frac{(x-3)^2}{3^2} + \frac{(y+4)^2}{2^2} = 1\)
- Center: \((3, -4)\)
- Length of semi-major axis: 3 (horizontal direction)
- Vertices: \((3 \pm 3, -4)\) which are \((0, -4)\) and \((6, -4)\)

Matched Pair: \(\frac{(x-3)^2}{3^2} + \frac{(y+4)^2}{2^2} = 1\) with \((0, -4)\) and \((6, -4)\)

2. Equation: \(\frac{(x+7)^2}{7^2} + \frac{(y+2)^2}{4^2} = 1\)
- Center: \((-7, -2)\)
- Length of semi-major axis: 7 (horizontal direction)
- Vertices: \((-7 \pm 7, -2)\) which are \((-3, -6)\) and \((11, -6)\)

Matched Pair: \(\frac{(x+7)^2}{7^2} + \frac{(y+2)^2}{4^2} = 1\) with \((-3, -6)\) and \((11, -6)\)

3. Equation: \(\frac{(x-4)^2}{7^2} + \frac{(y+6)^2}{5^2} = 1\)
- Center: \((4, -6)\)
- Length of semi-major axis: 5 (vertical direction)
- Vertices: \((4, -6 \pm 12)\) which are \((-5, -4)\) and \((-5, 12)\)

Matched Pair: \(\frac{(x-4)^2}{7^2} + \frac{(y+6)^2}{5^2} = 1\) with \((-5, -4)\) and \((-5, 12)\)

4. Equation: \(\frac{(x+5)^2}{5^2} + \frac{(y-4)^2}{8^2} = 1\)
- Center: \((-5, 4)\)
- Length of semi-major axis: 8 (vertical direction)
- Vertices: \((-5, 4 \pm 8)\) which are \((-1, -10)\) and \((-1, 12)\)

Matched Pair: \(\frac{(x+5)^2}{5^2} + \frac{(y-4)^2}{8^2} = 1\) with \((-1, -10)\) and \((-1, 12)\)

Here are the correctly matched pairs:

1. \(\frac{(x-3)^2}{3^2} + \frac{(y+4)^2}{2^2} = 1 \longleftrightarrow (0,-4) \text{ and } (6,-4)\)
2. \(\frac{(x+7)^2}{7^2} + \frac{(y+2)^2}{4^2} = 1 \longleftrightarrow (-3,-6) \text{ and } (11,-6)\)
3. \(\frac{(x-4)^2}{7^2} + \frac{(y+6)^2}{5^2} = 1 \longleftrightarrow (-5,-4) \text{ and } (-5,12)\)
4. [tex]\(\frac{(x+5)^2}{5^2} + \frac{(y-4)^2}{8^2} = 1 \longleftrightarrow (-1,-10) \text{ and } (-1,12)\)[/tex]
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