Consider the following for the next two (02) items that follow : P(x, y) is any point on the ellipse x2 + 4y2 = 1. Let E, F be the foci of the ellipse.
What is PE + PF equal to ?
2
Understanding the Ellipse Property: Sum of Distances to Foci
The question asks us to find the sum of the distances from any point P(x, y) on the ellipse \(x^2 + 4y^2 = 1\) to its two foci, denoted as E and F. This sum is a fundamental property of an ellipse.
Analyzing the Given Ellipse Equation
The equation of the ellipse is given as \(x^2 + 4y^2 = 1\). To understand its properties, we should convert this equation into the standard form of an ellipse centered at the origin, which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Dividing the given equation \(x^2 + 4y^2 = 1\) by 1 (which doesn't change the equation), we get:
\[ \frac{x^2}{1} + \frac{4y^2}{1} = 1 \]To get the \(y^2\) term in the standard form \(\frac{y^2}{b^2}\), we rewrite \(\frac{4y^2}{1}\) as \(\frac{y^2}{1/4}\):
\[ \frac{x^2}{1} + \frac{y^2}{1/4} = 1 \]Now, comparing this with the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify \(a^2\) and \(b^2\):
- \(a^2 = 1\)
- \(b^2 = 1/4\)
Since \(a^2 > b^2\) (\(1 > 1/4\)), the major axis of the ellipse lies along the x-axis. The length of the semi-major axis is \(a = \sqrt{1} = 1\). The length of the semi-minor axis is \(b = \sqrt{1/4} = 1/2\).
The Key Property of an Ellipse
One of the defining properties of an ellipse is that for any point P on the ellipse, the sum of the distances from P to the two foci (PE + PF) is constant and is equal to the length of the major axis. The length of the major axis is \(2a\).
In our case, the semi-major axis length is \(a = 1\). Therefore, the length of the major axis is \(2a = 2 \times 1 = 2\).
According to the property, for any point P on the ellipse \(x^2 + 4y^2 = 1\), the sum of the distances to the foci E and F is:
\[ PE + PF = 2a \] \[ PE + PF = 2 \times 1 \] \[ PE + PF = 2 \]Thus, the sum of the distances PE + PF is equal to 2.
| Property | Value | Calculation |
|---|---|---|
| Standard Form | \(\frac{x^2}{1} + \frac{y^2}{1/4} = 1\) | \(x^2 + 4y^2 = 1 \implies \frac{x^2}{1} + \frac{y^2}{1/4} = 1\) |
| \(a^2\) | 1 | From standard form |
| \(b^2\) | 1/4 | From standard form |
| Semi-major axis (a) | 1 | \(\sqrt{a^2} = \sqrt{1} = 1\) |
| Semi-minor axis (b) | 1/2 | \(\sqrt{b^2} = \sqrt{1/4} = 1/2\) |
| Major axis orientation | Along x-axis | Since \(a^2 > b^2\) |
| Length of Major Axis (2a) | 2 | \(2 \times 1 = 2\) |
| Sum of distances to foci (PE + PF) | 2a | Property of ellipse |
Conclusion
For any point P on the ellipse \(x^2 + 4y^2 = 1\), the sum of its distances from the two foci E and F is equal to the length of the major axis, which is 2.
Revision Table: Ellipse Foci and Distances
| Concept | Description | Relevance to Question |
|---|---|---|
| Ellipse Definition | Set of all points P such that the sum of distances from P to two fixed points (foci) is constant. | This constant sum is what we need to find. |
| Standard Equation (\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)) | Equation for an ellipse centered at origin with major axis along x-axis (if \(a > b\)) or y-axis (if \(b > a\)). | Used to find 'a' for the given equation. |
| Foci (E, F) | The two fixed points used in the definition of an ellipse. Located on the major axis. | PE and PF are distances to these points. |
| Major Axis | The longest diameter of the ellipse, passing through the foci. Length is 2a. | The sum of distances PE + PF equals the length of the major axis (2a). |
Additional Information: Ellipse Properties
Ellipses are fascinating conic sections with several important properties. Beyond the sum of distances to foci, here are a few more key concepts related to ellipses:
- Minor Axis: The shortest diameter of the ellipse, perpendicular to the major axis. Its length is 2b.
- Center: The midpoint of both the major and minor axes. In the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the center is at (0,0).
- Vertices: The endpoints of the major axis. For an ellipse with the major axis on the x-axis, the vertices are at \((\pm a, 0)\).
- Co-vertices: The endpoints of the minor axis. For an ellipse with the minor axis on the y-axis, the co-vertices are at \((0, \pm b)\).
- Eccentricity (e): A measure of how 'squashed' the ellipse is. Defined as \(e = c/a\), where c is the distance from the center to a focus (\(c^2 = a^2 - b^2\)). For an ellipse, \(0 < e < 1\). A circle is a special case of an ellipse with \(e = 0\) (where foci coincide at the center).
- Latus Rectum: A chord through a focus perpendicular to the major axis. The length of each latus rectum is \(2b^2/a\).
These properties help fully describe the shape and orientation of an ellipse.
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