Chapter Twenty-Two — Conic Sections
The Humongous Book of Algebra Problems
493
Multiply the entire equation by –1 and solve for x.
The equation is in standard form x = a(y – k)
2
+ h; therefore, a = 2, , and
. Calculate c.
Because a > 0, the focus is right of the vertex: (h + c,k).
The directrix is the vertical line c units left of the vertex: x = h – c.
The focus of the parabola is and the directrix is .
22.11 Write the equation of the parabola with vertex and directrix x = 0 in
standard form.
If the vertex is , then and k = 0. The directrix is a vertical line
one half of a unit right of the vertex, so the focus is the point one half of a unit
left of the vertex along the axis of symmetry: (–1,0). Note that c represents
the distance between the vertex and either the directrix or the focus—the
distances are equivalent—so .
Substitute into the formula to calculate a.
This parab-
ola contains y
2
,
so the rules about
the focus and
directrix in Problems
22.8 and 22.9 have to
be adjusted. Change
“above” to “right of”
and “below” to “left of.”
Instead of adding c to
the y-value of the
focus, add it to
the x-value.
The ends of
this parabola point
left. Its directrix is a
vertical line, so that tells
you it opens either right or
left. The axis of symmetry
is the horizontal line passing
through the vertex:
y = 0.
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