Math  /  Geometry

QuestionWrite the equation of the directrix of the parabola shown below. Write your answer without using spaces. y24x+4y4=0y^{2}-4 x+4 y-4=0

Studdy Solution

STEP 1

What is this asking? We need to find the line that helps define this parabola, called the directrix! Watch out! Don't mix up the directrix and the focus; they're related but on opposite sides of the parabola.
Also, remember the standard form of a parabola equation to make things easier!

STEP 2

1. Rewrite the equation
2. Find the vertex
3. Find the focal length
4. Find the axis of symmetry
5. Determine the directrix

STEP 3

Let's **group** our yy terms: y2+4y4x4=0y^2 + 4y - 4x - 4 = 0.
To **complete the square**, we take half of the coefficient of our yy term (which is **4**, half of that is **2**), square it (**2** squared is **4**), and add it to both sides: y2+4y+44x4=4y^2 + 4y + \mathbf{4} - 4x - 4 = \mathbf{4}.

STEP 4

This simplifies to (y+2)24x=8(y+2)^2 - 4x = 8, which we can rewrite as (y+2)2=4x+8(y+2)^2 = 4x + 8.
Factoring out a 4\mathbf{4} on the right side gives us (y+2)2=4(x+2)(y+2)^2 = 4(x+2).
This looks much closer to our standard form!

STEP 5

Our equation is now in the form (yk)2=4p(xh)(y-k)^2 = 4p(x-h), where (h,k)(h,k) is the **vertex**.
Matching this with our equation, we see h=2h = -2 and k=2k = -2.
So, our **vertex** is at (2,2)(-2,-2)!

STEP 6

In our standard form, 4p4p is equal to the coefficient of the term with xx.
In our equation, that's 4\mathbf{4}.
So, 4p=44p = 4, which means p=1p = \mathbf{1}.
This p\mathbf{p} is the **focal length**, the distance from the vertex to the focus (and also to the directrix).

STEP 7

Since our equation is in the form (yk)2=4p(xh)(y-k)^2 = 4p(x-h), it opens either to the right or left.
Because our pp value (1\mathbf{1}) is **positive**, it opens to the *right*.
This means the axis of symmetry is a horizontal line through the vertex, so it's y=2y = -2.

STEP 8

The directrix is a vertical line pp units to the *left* of the vertex (since the parabola opens to the *right*).
Our vertex is at (2,2)(-2,-2) and p=1p = \mathbf{1}.
Moving 1\mathbf{1} unit to the left means subtracting 1\mathbf{1} from the xx-coordinate of the vertex.
So, the directrix is x=21x = -2 - \mathbf{1}, which simplifies to x=3x = \mathbf{-3}.

STEP 9

The equation of the directrix is x=3x=-3.

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