What is the distance between the foci of the ellipse x 2+ 2y 2= 1 ?
Finding the Distance Between Foci of the Ellipse \(x^2 + 2y^2 = 1\)
Let's find the distance between the foci of the given ellipse equation. The equation of the ellipse is provided as:
\[x^2 + 2y^2 = 1\]To work with this equation, we first need to convert it into the standard form of an ellipse equation, which is generally given by:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]or
\[\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\]The standard form depends on whether the major axis is along the x-axis or the y-axis. In the standard form, \(a^2\) is always the larger denominator.
Comparing our given equation with the standard form, we can rewrite it as:
\[\frac{x^2}{1} + \frac{y^2}{1/2} = 1\]Here, the denominators are 1 and 1/2. Since 1 > 1/2, the larger denominator is \(a^2\), and the smaller denominator is \(b^2\).
- \(a^2 = 1 \implies a = \sqrt{1} = 1\)
- \(b^2 = \frac{1}{2} \implies b = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}\)
Since \(a^2\) is under the \(x^2\) term, the major axis of the ellipse is along the x-axis. The foci of an ellipse with the major axis along the x-axis are located at \((\pm c, 0)\), where \(c\) is the distance from the center \((0,0)\) to each focus.
The relationship between \(a\), \(b\), and \(c\) for an ellipse is given by the equation:
\[c^2 = a^2 - b^2\]Now, we can substitute the values of \(a^2\) and \(b^2\) into this equation to find \(c^2\):
\[c^2 = 1 - \frac{1}{2}\] \[c^2 = \frac{2}{2} - \frac{1}{2}\] \[c^2 = \frac{1}{2}\]To find \(c\), we take the square root of \(c^2\):
\[c = \sqrt{\frac{1}{2}}\] \[c = \frac{\sqrt{1}}{\sqrt{2}}\] \[c = \frac{1}{\sqrt{2}}\]We can rationalize the denominator:
\[c = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]The foci are located at \((\pm c, 0)\), which are \((\frac{\sqrt{2}}{2}, 0)\) and \((-\frac{\sqrt{2}}{2}, 0)\).
The distance between the two foci is \(2c\).
Distance between foci \( = 2 \times c\)
Substituting the value of \(c\):
Distance between foci \( = 2 \times \frac{1}{\sqrt{2}}\)
To simplify this, we can write \(2\) as \(\sqrt{2} \times \sqrt{2}\):
Distance between foci \( = \frac{\sqrt{2} \times \sqrt{2}}{\sqrt{2}}\)
Cancel out one \(\sqrt{2}\) from the numerator and denominator:
Distance between foci \( = \sqrt{2}\)
Therefore, the distance between the foci of the ellipse \(x^2 + 2y^2 = 1\) is \(\sqrt{2}\).
Let's summarize the key values:
| Parameter | Value |
|---|---|
| Ellipse Equation | \(x^2 + 2y^2 = 1\) |
| Standard Form | \(\frac{x^2}{1} + \frac{y^2}{1/2} = 1\) |
| \(a^2\) | 1 |
| \(b^2\) | 1/2 |
| \(c^2 = a^2 - b^2\) | 1/2 |
| \(c\) | \(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\) |
| Distance between foci (\(2c\)) | \(\sqrt{2}\) |
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