What is an equation of a parabola with the given vertex and focus (-2 5) (-2 6).

Equation of parabola is
4
y
=
x
2
+
4
x
+
24
Explanation:
As the vertex
(
−
2
,
5
)
and focus
(
−
2
,
6
)
share same abscissa i.e.
−
2
, parabola has axis of symmetry as
x
=
−
2
or
x
+
2
=
0
Hence, equation of parabola is of the type
(
y
−
k
)
=
a
(
x
−
h
)
2
, where
(
h
,
k
)
is vertex. Its focus then is
(
h
,
k
+
1
4
a
)
As vertex is given to be
(
−
2
,
5
)
, the equation of parabola is
y
−
5
=
a
(
x
+
2
)
2
as vertex is
(
−
2
,
5
)
and parabola passes through vertex.
and its focus is
(
−
2
,
5
+
1
4
a
)
Therefore
5
+
1
4
a
=
6
or
1
4
a
=
1
i.e.
a
=
1
4
and equation of parabola is
y
−
5
=
1
4
(
x
+
2
)
2
or
4
y
−
20
=
(
x
+
2
)
2
=
x
2
+
4
x
+
4
or
4
y
=
x
2
+
4
x
+
24
graph{4y=x^2+4x+24 [-11.91, 8.09, -0.56, 9.44]}

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