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−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
4(x−6)=(y+3)2
4 left parenthesis x minus 6 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
16(x−6)=(y+3)2
16 left parenthesis x minus 6 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
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−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
4(x−6)=(y+3)2
4 left parenthesis x minus 6 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
16(x−6)=(y+3)2
16 left parenthesis x minus 6 right parenthesis equals left parenthesis y plus 3 right parenthesis squared
Answers
GPT-4o mini
To find the equation of the parabola with a focus at (10, -3) and a directrix of \( x = 2 \), let's first identify the key elements.
1. **Focus**: \( (10, -3) \)
2. **Directrix**: \( x = 2 \)
### Step 1: Determine the Vertex
The vertex of a parabola is located halfway between the focus and the directrix. The x-coordinate of the vertex can be calculated as:
\[
x_v = \frac{x_{\text{focus}} + x_{\text{directrix}}}{2} = \frac{10 + 2}{2} = 6
\]
The y-coordinate of the vertex will be the same as the y-coordinate of the focus, which is -3.
Thus, the vertex is at \( (6, -3) \).
### Step 2: Determine the Orientation and the Value of p
Since the focus is to the right of the directrix, the parabola opens to the right. The distance \( p \) (the distance from the vertex to the focus) can be calculated as:
\[
p = x_{\text{focus}} - x_v = 10 - 6 = 4
\]
### Step 3: Write the Equation of the Parabola
The standard form of the equation for a parabola that opens to the right is:
\[
(y - k)^2 = 4p(x - h)
\]
Where \( (h, k) \) is the vertex. Substituting in our vertex \( (6, -3) \) and \( p = 4 \):
\[
(y + 3)^2 = 4(4)(x - 6)
\]
Simplifying gives:
\[
(y + 3)^2 = 16(x - 6)
\]
So the correct equation of the parabola is:
\[
\boxed{16(x-6) = (y+3)^2}
\]
This matches the option provided:
**16(x−6)=(y+3)2**
1. **Focus**: \( (10, -3) \)
2. **Directrix**: \( x = 2 \)
### Step 1: Determine the Vertex
The vertex of a parabola is located halfway between the focus and the directrix. The x-coordinate of the vertex can be calculated as:
\[
x_v = \frac{x_{\text{focus}} + x_{\text{directrix}}}{2} = \frac{10 + 2}{2} = 6
\]
The y-coordinate of the vertex will be the same as the y-coordinate of the focus, which is -3.
Thus, the vertex is at \( (6, -3) \).
### Step 2: Determine the Orientation and the Value of p
Since the focus is to the right of the directrix, the parabola opens to the right. The distance \( p \) (the distance from the vertex to the focus) can be calculated as:
\[
p = x_{\text{focus}} - x_v = 10 - 6 = 4
\]
### Step 3: Write the Equation of the Parabola
The standard form of the equation for a parabola that opens to the right is:
\[
(y - k)^2 = 4p(x - h)
\]
Where \( (h, k) \) is the vertex. Substituting in our vertex \( (6, -3) \) and \( p = 4 \):
\[
(y + 3)^2 = 4(4)(x - 6)
\]
Simplifying gives:
\[
(y + 3)^2 = 16(x - 6)
\]
So the correct equation of the parabola is:
\[
\boxed{16(x-6) = (y+3)^2}
\]
This matches the option provided:
**16(x−6)=(y+3)2**