Problem 27
Question
Find the equation of the given central conic. Ellipse with foci \((\pm 2,0)\) and directrices \(x=\pm 8\)
Step-by-Step Solution
Verified Answer
The equation of the ellipse is \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \).
1Step 1: Understand the Elements of the Ellipse
For an ellipse centered at the origin, the foci \((\pm c, 0)\) suggest an ellipse aligned with the x-axis. Furthermore, the formulas for foci and directrix could be useful. - Foci: \(c=2\). - Directrices: \(x=\pm 8\) is related to \(a^2/c\).
2Step 2: Relate Given Values to Ellipse Properties
The ellipse's major axis is along the x-axis since the foci are \((\pm 2, 0)\). The distance from center to directrix \(x=a^2/c=8\). This results in equations \(c=2\) and \(a^2 = 8 \cdot 2 = 16\).
3Step 3: Solve for the Semi-Major Axis 'a'
We have \(a^2 = 16\). Therefore, \(a = \sqrt{16} = 4\). The semi-major axis length is \(a=4\).
4Step 4: Solve for the Semi-Minor Axis 'b'
Use the relationship \(b^2 = a^2 - c^2\) to find \(b\). Substituting \(a^2 = 16\) and \(c^2 = 4\), we get \(b^2 = 16 - 4 = 12\). Thus, \(b = \sqrt{12} = 2\sqrt{3}\).
5Step 5: Write the Standard Equation of the Ellipse
Since the ellipse is horizontal (foci on x-axis), use the standard form: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Plug in \(a^2 = 16\) and \(b^2 = 12\): \[ \frac{x^2}{16} + \frac{y^2}{12} = 1 \].
Key Concepts
Ellipse EquationFoci and DirectricesSemi-Major Axis and Semi-Minor Axis
Ellipse Equation
An ellipse is a fascinating geometrical shape that can be represented mathematically by an equation. When dealing with an ellipse centered at the origin, the equation takes on a standard form.
This standard equation is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Here, \(a\) represents the length of the semi-major axis and \(b\) is associated with the semi-minor axis length. The terms \(a^2\) and \(b^2\) are crucial in defining how stretched or round the ellipse appears.
If \(a > b\), the ellipse stretches more along the x-axis, and if \(b > a\), it stretches along the y-axis. This formula allows us to easily plot the ellipse and understand its physical dimensions.
In the given exercise, we've found that:
This standard equation is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Here, \(a\) represents the length of the semi-major axis and \(b\) is associated with the semi-minor axis length. The terms \(a^2\) and \(b^2\) are crucial in defining how stretched or round the ellipse appears.
If \(a > b\), the ellipse stretches more along the x-axis, and if \(b > a\), it stretches along the y-axis. This formula allows us to easily plot the ellipse and understand its physical dimensions.
In the given exercise, we've found that:
- \(a^2 = 16\)
- \(b^2 = 12\)
Foci and Directrices
The foci and directrices of an ellipse provide extra characteristics that help capture its essence.
In any ellipse aligned along the x-axis, the foci are essential points located inside the ellipse at a distance \(c\) from the center. These points are symmetric about the origin.
For the problem at hand, the foci are located at \((\pm 2, 0)\), indicating \(c = 2\). This means the ellipse stretches horizontally and its major axis sits on the x-axis.
Directrices are theoretical vertical lines that aid in the geometric explanation of the ellipse. The distance from the center of the ellipse to each directrix provides a way to relate \(a^2\), \(b^2\), and \(c^2\). For an ellipse, the formula is given by: \[ x = \pm \frac{a^2}{c} \]Using our exercise values, we derived the directrices as \(x = \pm 8\). This relationship is a reflection of the balance between the ellipse's horizontal stretch and its corresponding geometric properties.
In any ellipse aligned along the x-axis, the foci are essential points located inside the ellipse at a distance \(c\) from the center. These points are symmetric about the origin.
For the problem at hand, the foci are located at \((\pm 2, 0)\), indicating \(c = 2\). This means the ellipse stretches horizontally and its major axis sits on the x-axis.
Directrices are theoretical vertical lines that aid in the geometric explanation of the ellipse. The distance from the center of the ellipse to each directrix provides a way to relate \(a^2\), \(b^2\), and \(c^2\). For an ellipse, the formula is given by: \[ x = \pm \frac{a^2}{c} \]Using our exercise values, we derived the directrices as \(x = \pm 8\). This relationship is a reflection of the balance between the ellipse's horizontal stretch and its corresponding geometric properties.
Semi-Major Axis and Semi-Minor Axis
The semi-major and semi-minor axes are fundamental elements that define the size and shape of an ellipse.
The semi-major axis is the longest radius of the ellipse, running from the center to a point at the edge. Here, it's aligned horizontally, making this the x-axis for our ellipse.
From the exercise, we calculated the semi-major axis \(a\) to be:
We derived its length using the formula \(b^2 = a^2 - c^2\). Substituting our known values gives:
The semi-major axis is the longest radius of the ellipse, running from the center to a point at the edge. Here, it's aligned horizontally, making this the x-axis for our ellipse.
From the exercise, we calculated the semi-major axis \(a\) to be:
- \(a^2 = 16 \implies a = \sqrt{16} = 4\)
We derived its length using the formula \(b^2 = a^2 - c^2\). Substituting our known values gives:
- \(b^2 = 16 - 4 = 12 \implies b = \sqrt{12} = 2\sqrt{3}\)
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