Question
Write the equation of the parabola with a focus at
(
10
,
−
3
)
and a directrix of
x
=
2
. It will be helpful to draw a sketch first and identify the information needed to write the equation.(1 point)
Responses
4
(
x
−
6
)
=
(
y
+
3
)
2
4 left parenthesis x minus 6 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
4
(
x
−
10
)
=
(
y
+
3
)
2
4 left parenthesis x minus 10 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
(
x
−
6
)
2
=
16
(
y
+
3
)
left parenthesis x minus 6 right parenthesis squared equals 16 left parenthesis y plus 3 right parenthesis
16
(
x
−
6
)
=
(
y
+
3
)
2
(
10
,
−
3
)
and a directrix of
x
=
2
. It will be helpful to draw a sketch first and identify the information needed to write the equation.(1 point)
Responses
4
(
x
−
6
)
=
(
y
+
3
)
2
4 left parenthesis x minus 6 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
4
(
x
−
10
)
=
(
y
+
3
)
2
4 left parenthesis x minus 10 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
(
x
−
6
)
2
=
16
(
y
+
3
)
left parenthesis x minus 6 right parenthesis squared equals 16 left parenthesis y plus 3 right parenthesis
16
(
x
−
6
)
=
(
y
+
3
)
2
Answers
GPT-4o mini
To find the equation of the parabola with a focus at \( (10, -3) \) and a directrix of \( x = 2 \), we first determine the vertex and the orientation of the parabola.
1. **Determine the Vertex**:
The vertex of a parabola is located halfway between the focus and the directrix. The focus is at \( (10, -3) \) and the directrix is the vertical line \(x = 2\).
- The x-coordinate of the focus is 10, and the respective x-coordinate of the directrix is 2.
- The midpoint, or vertex \( (h, k) \), can be calculated as:
\[
h = \frac{10 + 2}{2} = 6, \quad k = -3
\]
Thus, the vertex is located at \( (6, -3) \).
2. **Orientation**:
Since the focus is to the right of the directrix, the parabola opens to the right.
3. **Standard Form of the Equation**:
The standard form of a horizontally oriented parabola is given by:
\[
(y - k)^2 = 4p(x - h)
\]
Where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus.
4. **Calculating \( p \)**:
The distance \( p \) can be found as the distance from the vertex \( (6, -3) \) to the focus \( (10, -3) \):
\[
p = 10 - 6 = 4
\]
5. **Substituting into the Equation**:
With \( h = 6 \), \( k = -3 \), and \( p = 4 \), we substitute into the standard form:
\[
(y + 3)^2 = 4 \cdot 4 (x - 6)
\]
This simplifies to:
\[
(y + 3)^2 = 16(x - 6)
\]
Thus, the equation of the parabola is:
\[
\boxed{(y + 3)^2 = 16(x - 6)}
\]
1. **Determine the Vertex**:
The vertex of a parabola is located halfway between the focus and the directrix. The focus is at \( (10, -3) \) and the directrix is the vertical line \(x = 2\).
- The x-coordinate of the focus is 10, and the respective x-coordinate of the directrix is 2.
- The midpoint, or vertex \( (h, k) \), can be calculated as:
\[
h = \frac{10 + 2}{2} = 6, \quad k = -3
\]
Thus, the vertex is located at \( (6, -3) \).
2. **Orientation**:
Since the focus is to the right of the directrix, the parabola opens to the right.
3. **Standard Form of the Equation**:
The standard form of a horizontally oriented parabola is given by:
\[
(y - k)^2 = 4p(x - h)
\]
Where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus.
4. **Calculating \( p \)**:
The distance \( p \) can be found as the distance from the vertex \( (6, -3) \) to the focus \( (10, -3) \):
\[
p = 10 - 6 = 4
\]
5. **Substituting into the Equation**:
With \( h = 6 \), \( k = -3 \), and \( p = 4 \), we substitute into the standard form:
\[
(y + 3)^2 = 4 \cdot 4 (x - 6)
\]
This simplifies to:
\[
(y + 3)^2 = 16(x - 6)
\]
Thus, the equation of the parabola is:
\[
\boxed{(y + 3)^2 = 16(x - 6)}
\]