Answered

IDNLearn.com provides a seamless experience for finding and sharing answers. Our platform provides detailed and accurate responses from experts, helping you navigate any topic with confidence.

Find the equation for the parabola with the following focus and directrix:

Focus: [tex]\((-2,1)\)[/tex]

Directrix: [tex]\(x=-8\)[/tex]

[tex]\[
(y - \square)^2 = \square(x + \square)
\][/tex]

Enter the correct values in the squares.

Sagot :

GinnyAnsweridn
To find the equation of the parabola given the focus and directrix, follow these steps:

1. Identify known parts of the parabola:
- Focus: [tex]\( (-2, 1) \)[/tex]
- Directrix: [tex]\( x = -8 \)[/tex]

2. Determine the key components:
- The vertex form of a parabola with a vertical directrix is: [tex]\( (y - k)^2 = 4p(x - h) \)[/tex]
- Here, [tex]\((h, k)\)[/tex] represents the focus of the parabola.
- Given the directrix equation [tex]\( x = -8 \)[/tex], the relationship with the focus is given by [tex]\( x = h - p \)[/tex].

3. Substitute the known value of the focus:
- From the given focus, [tex]\( h = -2 \)[/tex] and [tex]\( k = 1 \)[/tex].

4. Find the value of [tex]\( p \)[/tex]:
- The directrix [tex]\( x = -8 \)[/tex] implies that:
[tex]\( -8 = -2 - p \)[/tex]

- Solving for [tex]\( p \)[/tex]:
[tex]\[ -8 = -2 - p \][/tex]
[tex]\[ p = -6 \][/tex]

5. Write the equation of the parabola:
- Substitute [tex]\( h = -2 \)[/tex], [tex]\( k = 1 \)[/tex], and [tex]\( p = -6 \)[/tex] into the vertex form equation:
[tex]\[ (y - 1)^2 = 4p(x - (-2)) \][/tex]

6. Simplify the equation:
- Substituting [tex]\( p = -6 \)[/tex] gives:
[tex]\[ (y - 1)^2 = 4(-6)(x - (-2)) \][/tex]
[tex]\[ (y - 1)^2 = -24(x + 2) \][/tex]

So, the equation of the parabola is: [tex]\((y - 1)^2 = -24(x + 2)\)[/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com is committed to providing accurate answers. Thanks for stopping by, and see you next time for more solutions.