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)

Question

Responses

4(x−6)=(y+3)2
4 left parenthesis x minus 6 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−10)=(y+3)2
4 left parenthesis x minus 10 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

Answer 1

To find the equation of a parabola given a focus and a directrix, we can follow these steps:

1. Identify the Focus and Directrix:

   • Focus: \( (10, -3) \)
   • Directrix: \( x = 2 \)

2. Determine the Orientation: Since the focus is to the right of the directrix, the parabola opens to the right.

3. Find the Vertex: The vertex lies midway between the focus and the directrix. To find the x-coordinate of the vertex:

   • The x-coordinates of the focus and the directrix are \(10\) and \(2\), respectively.
   • The x-coordinate of the vertex is: \[ \frac{10 + 2}{2} = \frac{12}{2} = 6 \] The y-coordinate remains the same as the focus \( -3 \).
   • Vertex: \( (6, -3) \)

4. Find the value of \(p\): The distance \(p\) from the vertex to the focus (or vertex to directrix) is: \[ p = 10 - 6 = 4 \]

5. Write the Equation: The standard form of a parabola that opens to the right, with vertex \((h, k)\) is: \[ (y - k)^2 = 4p(x - h) \] Here:

   • \(h = 6\)
   • \(k = -3\)
   • \(p = 4\)

   Substituting in these values: \[ (y + 3)^2 = 16(x - 6) \]

Thus, the equation of the parabola is: \[ (y + 3)^2 = 16(x - 6) \]

Among the options you provided, this equation corresponds to: \[ \text{(x−6)²=16(y+3)} \] This represents the correct form of the equation of the parabola.