To Show: The parabola with the given vertex and the given focus has the given equation, using the definition of a parabola.
It has been shown using the definition of a parabola, that the parabola with the vertex,
Given:
The vertex,
Concept used:
A parabola is defined as the locus of a point which is equidistant from a point called focus and a line called the directrix.
Calculation:
As discussed, a parabola is defined as the locus of a point which is equidistant from a point called focus and a line called the directrix.
It is given that the parabola has its vertex at
Note that these points both lie on the line
Then, the directrix, which is perpendicular to the axis of symmetry, must be a horizontal line of the form
The foot of the perpendicular drawn from the vertex,
Since the vertex lies on the parabola, it must be equidistant from the focus and directrix.
The distance between the vertex,
The distance between the vertex,
According to the problem,
Simplifying,
So, the directrix is given as
Now, let
According to the definition, it is equidistant from the focus and the directrix.
The distance between
Now, the distance between
Applying distance formula, the distance between
According to the problem,
Squaring both sides,
Simplifying,
On further simplification,
Finally,
Since
This is the required proof.
Conclusion:
It has been shown using the definition of a parabola, that the parabola with the vertex,
Chapter 10 Solutions
High School Math 2015 Common Core Algebra 2 Student Edition Grades 10/11
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