The equation of the directrix of the parabola y² + 4y + 4x + 2 = 0 is (a) x = -1 (b) x = 1 (c) x = -

Q: Wait, why do we complete the square? Can't we just use the general formula for a parabola?

Great question! While you could memorize a complex formula for the general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, it is often prone to errors. Completing the square is a fundamental technique that transforms the equation into the vertex form $(y-k)^2 = 4a(x-h)$, which makes identifying the vertex and the directrix almost instantaneous. It is safer and builds a deeper understanding of the geometry.

Q: Why is the directrix $X = a$ and not $X = -a$?

Excellent observation. The standard form $Y^2 = -4aX$ represents a parabola opening to the left. In this orientation, the focus is at $(-a, 0)$ relative to the vertex, and the directrix is a vertical line located at $X = a$ relative to the vertex. If it were a right-opening parabola ($Y^2 = 4aX$), the directrix would be at $X = -a$. Always sketch a quick graph to confirm the direction!

Q: I got $x = 1/2$ as my answer, but the options say $x = 3/2$. Where did I go wrong?

It sounds like you might have forgotten to add the distance $a$ to the vertex's $x$-coordinate. The directrix is not just the value of $a$; it is the line $x = h + a$. Since your vertex $h$ is $1/2$ and your $a$ is $1$, the directrix is $x = 1/2 + 1 = 3/2$. Don't worry, this is a very common mistake—just remember that the directrix is a line, not just a single value!

Question

The equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$ is

Options:

(A) $x = -1$

(B) $x = 1$

(C) $x = -3/2$

(D) $x = 3/2$

Sigma Percentile · Accepted Answer

### The Geometry of the Parabola: A Journey into Symmetry

My dear student, welcome to the beautiful world of coordinate geometry. Today, we are going to peel back the layers of a seemingly simple equation: $y^2 + 4y + 4x + 2 = 0$.

At first glance, it looks like a jumble of terms, but beneath this chaos lies a perfect, elegant symmetry. A parabola is not just an equation; it is a locus of points, a dance between a focus and a directrix. Let us uncover its secrets.

### Phase 1: The Algebraic Fog

When we look at $y^2 + 4y + 4x + 2 = 0$, we see a $y^2$ term and a $y$ term, but only a linear $x$ term. This is our first clue.

It tells us that this parabola is not vertical; it is horizontal, opening either to the left or the right. To understand its properties, we must bring it into the light of the standard form.

We start by grouping the $y$ terms: $(y^2 + 4y) + 4x + 2 = 0$. This is the first step in our journey—organizing the chaos.

### Phase 2: The Art of Completing the Square

Now, we must perform the magic of completing the square. We take the coefficient of $y$, which is $4$. We cut it in half to get $2$, and then we square that $2$ to get $4$.

To keep our equation balanced, we add and subtract $4$:
$$(y^2 + 4y + 4) - 4 + 4x + 2 = 0$$

Notice how the first three terms, $(y^2 + 4y + 4)$, collapse beautifully into the perfect square $(y+2)^2$. Our equation now reads $(y+2)^2 - 4 + 4x + 2 = 0$.

Simplifying the constants, we get $(y+2)^2 + 4x - 2 = 0$. This is the moment where the structure begins to emerge.

### Phase 3: The Symmetry Revealed

We are almost there. Let us isolate the squared term: $(y+2)^2 = -4x + 2$.

To match the standard form $Y^2 = -4aX$, we must factor out the coefficient of $x$. We pull out $-4$, giving us:
$$(y+2)^2 = -4\left(x - \frac{1}{2}\right)$$

Here, we define our new coordinates: $Y = y + 2$ and $X = x - \frac{1}{2}$. This tells us that the vertex of our parabola is at $(\frac{1}{2}, -2)$.

Because of the negative sign, we know this parabola opens to the left. It is a curve that reaches out into the negative $x$-direction.

### Phase 4: The Geometric Truth

Finally, we identify the parameter $a$. Comparing our equation to $Y^2 = -4aX$, we see that $4a = 4$, which means $a = 1$.

This $a$ is the distance from the vertex to the focus and, crucially, the distance from the vertex to the directrix. For a left-opening parabola, the directrix is a vertical line located at $X = a$.

Substituting our coordinate transformation back, we have $x - \frac{1}{2} = 1$. Solving for $x$, we find:
$$x = 1 + \frac{1}{2} = \frac{3}{2}$$

And there it is! The directrix is the line **$x = \frac{3}{2}$**.

It is the invisible fence that defines the parabola's shape. I hope you can see the elegance in this process. We started with a messy equation and, through careful algebraic steps, revealed the geometric truth hidden within.

Animated Solution for Mathematics - Conic Sections: The equation of the directrix of the parabola $y^{2} + 4 y + 4 x + 2 = 0$ is

Select Answer:

Visualized Solution

The Given Parabola

Grouping Terms

Completing the Square

Forming the Perfect Square

Isolating the Squared Term

Factoring the Right Side

Identifying the Standard Form

Locating the Vertex

Finding the Parameter $a$

The Directrix Formula

Substituting the Values

Final Calculation

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

The Geometry of the Parabola

Phase 1

Phase 2

Phase 3

Phase 4

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Animated Solution for Mathematics - Conic Sections: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the directrix of the parabola y2+4y+4x+2=0 is