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Concept explainers
In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand.
Vertex at ; directrix the line
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a & b
The equation of the parabola described and two points that define the latus rectum for the following description.
Focus at ; directrix the line
Answer to Problem 33AYU
Latus rectum: and .
Explanation of Solution
Given:
Focus at ; directrix the line
Formula used:
Vertex | Focus | Directrix | Equation | Description |
Parabola, axis of symmetry is parallel to , opens up |
Calculation:
The focus is at and the directrix the line . Since , axis of symmetry is parallel to . Focus at and directrix the line implies,
The Vertex is at ( coordinate of is 3 because the distance between vertex and focus is and hence
Focus lies above the Vertex. Hence the parabola opens up.
Hence the equation of parabola is,
By substituting
The equation of the parabola is,
Directrix D:
Substituting we get,
Hence and defines the latus rectum.
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c.
To graph: The parabola for the equation
Answer to Problem 33AYU
Explanation of Solution
Given:
Focus at ; directrix the line
Calculation:
The graph of the parabola
Using the given information Focus at ; directrix the line , we find Vertex and we have formed the equation and plotted the graph for
The focus and the vertex lie on the vertical line as shown. The equation for directrix D: is shown as dotted line. The parabola opens up.
The points and defines the latus rectum.
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