write the equation for a parabola with a focus at (2,2) and a directrix at x=8
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Step-by-step explanation:
The key to solving this problem is knowing all your relevant equations. The formula for a parabola in vertex form is:
(
y
−
k
)
2
=
4
p
(
x
−
h
)
Vertex:
(
h
,
k
)
Focus:
(
p
+
h
,
k
)
Directrix:
x
=
−
p
+
h
Notice we chose the version that squares the
y
-value. This is because the given directrix is
x
=
−
2
. If the directrix had been
y
=
−
2
, then the formulas are different.
You are given the focus of
(
2
,
2
)
(
h
,
k
)
=
(
2
,
2
)
So
h
=
2
and
k
=
2
You are also given the directrix
x
=
−
2
x
=
−
p
+
h
=
−
2
We already determined that
h
=
2
from above, so
−
p
+
2
=
−
2
−
p
=
−
4
(subtract 2 from both sides)
p
=
4
(divide both sides by -1)
Plugging these values back into the original equation gives
(
y
−
2
)
2
=
4
(
4
)
(
x
−
2
)
(
y
−
2
)
2
=
16
(
x
−
2
)
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