Problems based on finding focus, directrix, vertex, latus rectum etc.
Understanding the Ellipse: Focus, Directrix, Vertex, and Latus Rectum
An ellipse is a set of points in a plane such that the sum of the distances from two fixed points (foci) is constant. It is a conic section that can be thought of as a stretched circle. To solve problems involving ellipses, it is crucial to understand its key elements: focus, directrix, vertex, and latus rectum.
Key Elements of an Ellipse
Element | Description |
---|---|
Focus (Foci) | Two fixed points inside the ellipse. The sum of the distances from any point on the ellipse to the foci is constant. |
Directrix | A fixed line outside the ellipse. The ratio of the distance of any point on the ellipse from a focus to its distance from the corresponding directrix is constant (eccentricity). |
Vertex | The points where the ellipse is widest (major vertices) or narrowest (minor vertices). |
Latus Rectum | A line segment perpendicular to the major axis at a focus, with endpoints on the ellipse. Its length is directly related to the distance between the foci. |
Standard Form of an Ellipse
The standard form of an ellipse with its center at the origin (0, 0) is given by:
x2a2+y2b2=1
where:
- a is the semi-major axis length (distance from the center to a vertex on the major axis).
- b is the semi-minor axis length (distance from the center to a vertex on the minor axis).
Important Formulas
- Eccentricity (e): e=√1−b2a2 for a>b.
- Focus (Foci): (±ae,0) for a horizontal ellipse, or (0,±ae) for a vertical ellipse.
- Directrix: x=±ae for a horizontal ellipse, or y=±ae for a vertical ellipse.
- Vertex (Vertices): (±a,0) for a horizontal ellipse, or (0,±a) for a vertical ellipse.
- Latus Rectum Length (L): L=2b2a for a horizontal ellipse, or L=2a2b for a vertical ellipse.
Examples
Example 1: Finding the Focus and Directrix
Given the ellipse x216+y29=1, find the foci and directrices.
Solution:
- Identify a2 and b2. Here, a2=16 and b2=9.
- Calculate a and b. a=4 and b=3.
- Calculate the eccentricity e. e=√1−b2a2=√1−916=√716=√74
- Find the foci using the eccentricity. Foci=(±ae,0)=(±4⋅√74,0)=(±√7,0)
- Find the directrices. Directrix=x=±ae=±4√74=±16√7
Example 2: Finding the Latus Rectum
Given the same ellipse x216+y29=1, find the length of the latus rectum.
Solution:
- Use the formula for the latus rectum length for a horizontal ellipse. L=2b2a=2⋅94=184=4.5
- The length of the latus rectum is 4.5 units.
By understanding these elements and formulas, you can solve a wide range of problems related to ellipses. Practice with different equations of ellipses to become proficient in identifying and calculating the focus, directrix, vertices, and latus rectum.