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.
Consider the following points : 1. \(\left(\frac{\sqrt{3}}{2}, 0\right)\) 2. \(\left(\frac{\sqrt{3}}{2}, \frac{1}{4}\right)\) 3. \(\left(\frac{\sqrt{3}}{2},-\frac{1}{4}\right)\) Which of the above points lie on latus rectum of ellipse ?
1, 2 and 3
Understanding the Ellipse and its Latus Rectum
The question asks us to identify which of the given points lie on the latus rectum of the ellipse with the equation \(x^2 + 4y^2 = 1\). To solve this, we first need to understand the properties of the ellipse and what its latus rectum is.
Standard Form of the Ellipse Equation
The given equation of the ellipse is \(x^2 + 4y^2 = 1\). To work with this, we convert it into the standard form for an ellipse centered at the origin, which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Dividing the given equation by 1, we get:
\[ \frac{x^2}{1} + \frac{4y^2}{1} = 1 \] \[ \frac{x^2}{1} + \frac{y^2}{1/4} = 1 \] Comparing this with the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
\(a^2 = 1 \Rightarrow a = 1\) (since \(a > 0\))
\(b^2 = 1/4 \Rightarrow b = 1/2\) (since \(b > 0\))
Finding the Foci and Eccentricity
Since \(a^2 > b^2\) (\(1 > 1/4\)), the major axis of the ellipse lies along the x-axis. The foci of the ellipse are located at \((\pm ae, 0)\), where \(e\) is the eccentricity. The relationship between \(a\), \(b\), and \(e\) for an ellipse with the major axis along the x-axis is \(b^2 = a^2(1 - e^2)\).
Substituting the values of \(a^2\) and \(b^2\):
\[ \frac{1}{4} = 1 (1 - e^2) \] \[ \frac{1}{4} = 1 - e^2 \] \[ e^2 = 1 - \frac{1}{4} = \frac{3}{4} \] \[ e = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \] The foci are at \((\pm ae, 0)\), which are \((\pm 1 \cdot \sqrt{3}/2, 0) = (\pm \sqrt{3}/2, 0)\). The foci are \(E\) and \(F\).
Understanding the Latus Rectum
The latus recta (plural of latus rectum) of an ellipse are line segments passing through the foci and perpendicular to the major axis, with endpoints on the ellipse. Since the major axis is along the x-axis, the latus recta are vertical lines passing through the foci \((\pm ae, 0)\).
The equations of the lines containing the latus recta are \(x = ae\) and \(x = -ae\).
In our case, \(ae = 1 \cdot \sqrt{3}/2 = \sqrt{3}/2\). So, the equations of the lines containing the latus recta are \(x = \sqrt{3}/2\) and \(x = -\sqrt{3}/2\).
Checking the Given Points
We are given three points:
- \((\sqrt{3}/2, 0)\)
- \((\sqrt{3}/2, 1/4)\)
- \((\sqrt{3}/2, -1/4)\)
We need to check which of these points lie on the latus rectum of the ellipse. Based on the provided options and correct answer, it appears the question is asking which points lie on the lines containing the latus recta (specifically \(x = \pm ae\)), rather than strictly being endpoints or internal points of the latus rectum segment on the ellipse.
Let's examine the x-coordinate of each point:
- Point 1: \((\sqrt{3}/2, 0)\). The x-coordinate is \(\sqrt{3}/2\), which is equal to \(ae\). This point lies on the line \(x = \sqrt{3}/2\). This point is one of the foci.
- Point 2: \((\sqrt{3}/2, 1/4)\). The x-coordinate is \(\sqrt{3}/2\), which is equal to \(ae\). This point lies on the line \(x = \sqrt{3}/2\). Let's check if it's on the ellipse: \((\sqrt{3}/2)^2 + 4(1/4)^2 = 3/4 + 4(1/16) = 3/4 + 1/4 = 1\). Yes, it is on the ellipse. This is an endpoint of the latus rectum.
- Point 3: \((\sqrt{3}/2, -1/4)\). The x-coordinate is \(\sqrt{3}/2\), which is equal to \(ae\). This point lies on the line \(x = \sqrt{3}/2\). Let's check if it's on the ellipse: \((\sqrt{3}/2)^2 + 4(-1/4)^2 = 3/4 + 4(1/16) = 3/4 + 1/4 = 1\). Yes, it is on the ellipse. This is the other endpoint of the latus rectum on the positive x side.
All three points have an x-coordinate of \(\sqrt{3}/2\), which is equal to \(ae\). Therefore, all three points lie on the line \(x = \sqrt{3}/2\), which is one of the lines containing a latus rectum of the ellipse.
Conclusion
Based on the analysis, all three given points lie on the line \(x = \sqrt{3}/2\), which contains one of the latus recta of the ellipse \(x^2 + 4y^2 = 1\). Points 2 and 3 are the endpoints of this latus rectum segment on the ellipse, while point 1 is the focus.
Therefore, points 1, 2, and 3 lie on the latus rectum (interpreted as the line containing the latus rectum) of the ellipse.
| Point | Coordinates | X-coordinate | Lies on \(x = \sqrt{3}/2\)? | Lies on Ellipse? | Lies on Latus Rectum (Line)? |
|---|---|---|---|---|---|
| Point 1 | \((\sqrt{3}/2, 0)\) | \(\sqrt{3}/2\) | Yes | No (Focus) | Yes |
| Point 2 | \((\sqrt{3}/2, 1/4)\) | \(\sqrt{3}/2\) | Yes | Yes | Yes |
| Point 3 | \((\sqrt{3}/2, -1/4)\) | \(\sqrt{3}/2\) | Yes | Yes | Yes |
Revision Table: Ellipse Key Properties
| Property | Standard Form (\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(a > b\)) |
|---|---|
| Center | (0, 0) |
| Semi-major axis length | \(a\) |
| Semi-minor axis length | \(b\) |
| Foci | \((\pm ae, 0)\) |
| Eccentricity (\(e\)) | \(b^2 = a^2(1-e^2)\) or \(e = \sqrt{1 - \frac{b^2}{a^2}}\) |
| Equations of Latus Recta Lines | \(x = \pm ae\) |
| Length of Latus Rectum | \(\frac{2b^2}{a}\) |
| Endpoints of Latus Recta | \((\pm ae, \pm \frac{b^2}{a})\) |
Additional Information: Ellipse Concepts
An ellipse is a conic section formed by the intersection of a plane and a double-napped cone, where the plane is tilted such that it does not intersect the base of the cone and does not intersect the cone parallel to its axis. It is defined as the locus of all points \(P\) such that the sum of the distances from \(P\) to two fixed points (the foci, \(F_1\) and \(F_2\)) is constant, i.e., \(PF_1 + PF_2 = 2a\), where \(a\) is the length of the semi-major axis.
- Major Axis: The longest diameter of the ellipse, passing through the foci.
- Minor Axis: The shortest diameter of the ellipse, perpendicular to the major axis.
- Vertices: The endpoints of the major axis.
- Co-vertices: The endpoints of the minor axis.
- Eccentricity: A measure of how "stretched out" an ellipse is. It is the ratio of the distance from the center to a focus (\(ae\)) to the length of the semi-major axis (\(a\)), i.e., \(e = \frac{ae}{a}\). For an ellipse, \(0 < e < 1\).
- Latus Rectum: A chord of the ellipse perpendicular to the major axis and passing through a focus. There are two latus recta, one for each focus. Their length is \(\frac{2b^2}{a}\). The endpoints of the latus rectum through \((\sqrt{3}/2, 0)\) for our ellipse are \((\sqrt{3}/2, \pm b^2/a)\). Since \(b^2 = 1/4\) and \(a=1\), the endpoints are \((\sqrt{3}/2, \pm (1/4)/1) = (\sqrt{3}/2, \pm 1/4)\). These match points 2 and 3.
Understanding these key terms helps in solving problems related to ellipses and their properties.
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