MA 141 Lecture 2: Conic SectionsExam
Course CodeMA 141
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MA 141 Chapter 0
Section 0.2: Conic Sections
Distance Formula: (derived from Pythagorean Theorem) Given two points P= (x1, y1)
and Q= (x2, y2)in R2, the distance between them is
d(P, Q) = p(x1−x2)2+ (y1−y2)2.
Def: Aparabola is the set of points in the xy-plane that are equidistant from a ﬁxed point,
the focus, and a ﬁxed line, the directrix. The vertex is the point of the parabola that
minimizes the distance to the focus and directrix.
The standard equation of the vertical parabola is
In this case, the focus is h, k +1
4aand the directrix is y=k−1
If a > 0, the parabola opens upward and the vertex is the lowest point of the graph. If a < 0,
the parabola opens downward and the vertex is the highest point of the graph.
Similarly, we can have a parabola opening left or right. The standard equation in this
The focus is at h+1
4a, kand the directrix is x=h−1
If a > 0, the parabola opens to the right and the vertex is the left-most point. If a < 0, the
parabola opens left and the vertex is the right-most point.
Only pages 1-2 are available for preview. Some parts have been intentionally blurred.
Ex: Find the focus, vertex, and directrix of x=−1
6y2and then sketch the parabola.
This parabola is of the second type, opening either left or right. Since a=−1
it opens to the left.
This can be rewritten as x−0 = −1
6(y−0)2, so the vertex is at (0,0).
The focus is then given by 1
Then the focus is −3
2,0and the directrix is x=−−3
Note that if the parabola is not in the given form above, we can complete the square to
convert to the standard equation.
Recall: Given a quadratic y=x2+ 8x+ 3, we complete the square as follows.
y=x2+ 8x+ 3
= x2+ 8x+8
y+ 16 = (x2+ 8x+ 16) + 3
y+ 13 = (x+ 4)2
Then the standard form of the parabola is y+ 13 = (x+ 4)2.
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