Question

Find the vertex, focus, and Directrix of each parabola.
1. (x+2)^2=20(y+5)
2. (y-1)^2=-16(x+4)
3. (x-5)^2=4(y+1)
4.(y-3)^2=12(x+1)
5. (x+4)^2=6(y-5)

Answer 1

Answer to Question #91188 - Math - Analytic Geometry

Case-I

For a parabola of the shape as shown beside

The standard equation is

$$(x - h)^2 = 4p(y - k)$$

Vertex is

$$(h, k)$$

Focus is

$$(h, k + p)$$

And the equation of directrix is

$$y = k - p$$

Case-II

For a parabola of the shape shown beside

A standard equation of Parabola is

$$(y - a)^2 = 4p(x - b)$$

Vertex is

$$(b, a)$$

Focus is

$$(b + p, a)$$

And the equation of directrix is

$$x = b - p$$

(1). For the parabola, $\left(x + 2\right)^{2} = 20\left(y + 5\right)$

Find the vertex, focus, and Directrix of each parabola

**Solution**

The given equation of the parabola is

$$(x + 2)^2 = 20(y + 5) \quad \dots$$

This equation can be re-written as:

$$[x - (-2)]^2 = 4 \times 5[y - (-5)] \quad \dots$$

Comparing equation (2) with equation of parabola, $(x - h)^2 = 4p(y - k)$ ,

We have, $h = -2$ , $k = -5$ , $p = 5$ , therefore,

**Vertex**

$$(h, k) = (-2, -5)$$

**Focus** is

$$\begin{array}{l} (h, k + p) = (-2, -5 + 5) \\ = (-2, 0) \end{array}$$

And equation of directrix is $y = k - p = -5 - 5 \Rightarrow y = -10$

The given parabola, with its vertex, focus and the directrix is shown below

(3). For the parabola, $\left(x - 5\right)^{2} = 4\left(y + 1\right)$

Find the vertex, focus, and Directrix of each parabola

**Solution**

The given equation of the parabola is

$$(x - 5)^2 = 4(y + 1) \quad \dots$$

This equation can be re-written as:

$$[x - (5)]^2 = 4 \times 1[y - (-1)] \quad \dots$$

Comparing equation (2) with equation of parabola, $(x - h)^2 = 4p(y - k)$ ,

We have, $h = 5$ , $k = -1$ , $p = 1$ , therefore

**Vertex**

$$(h, k) = (5, -1)$$

**Focus** is

$$\begin{array}{l} (h, k + p) = (5, -1 + 1) \\ = (5, 0) \end{array}$$

And equation of directrix is $y = k - p = -1 - 1 \Rightarrow y = -2$

The given parabola, with its vertex, focus and the directrix is shown below

(5). For the parabola, $\left(x + 4\right)^{2} = 6\left(y - 5\right)$

Find the vertex, focus, and Directrix of each parabola

**Solution**

The given equation of the parabola is

$$(x + 4)^2 = 6(y - 5) \quad \dots$$

This equation can be re-written as:

$$[x - (-4)]^2 = 4 \times \frac{3}{2}[y - (5)] \quad \dots$$

Comparing equation (2) with equation of parabola, $(x - h)^2 = 4p(y - k)$ ,

We have, $h = -4$ , $k = 5$ , $p = \frac{3}{2}$ , therefore

**Vertex**

$$(h, k) = (-4, 5)$$

**Focus** is

$$(h, k + p) = \left(-4, 5 + \frac{3}{2}\right) = \left(-4, \frac{13}{2}\right)$$

And equation of directrix is

$$y = k - p = 5 - \frac{3}{2} \quad \Rightarrow \quad y = \frac{7}{2}$$

The given parabola, with its vertex, focus and the directrix is shown below

(2). For the parabola, $(y - 1)^2 = -16(x + 4)$

Find the vertex, focus, and Directrix of each parabola

**Solution**

The given equation of the parabola is

$$(y - 1)^2 = -16(x + 4) \quad \dots$$

This equation can be re-written as:

$$[y - (1)]^2 = 4 \times (-4)[x - (-4)] \quad \dots$$

Comparing equation (2) with equation of parabola, $(y - a)^2 = 4p(x - b)$

We have, $b = -4$, $a = 1$, $p = -4$, therefore

**Vertex**

$(b, a) = (-4, 1)$

**Focus** is

$(b + p, a) = (-4 - 4, 1) = (-8, 1)$

And equation of directrix is

$$x = b - p = -4 - (-4) \Rightarrow x = 0$$

The given parabola, with its vertex, focus and the directrix is shown below

(4). For the parabola, $\left(y - 3\right)^{2} = 12\left(x + 1\right)$

Find the vertex, focus, and Directrix of each parabola

**Solution**

The given equation of the parabola is

$$(y - 3)^2 = 12(x + 1) \quad \dots$$

This equation can be re-written as:

$$[y - (3)]^2 = 4 \times (3)[x - (-1)] \quad \dots$$

Comparing equation (2) with equation of parabola, $(y - a)^2 = 4p(x - b)$ ,

We have, $b = -1$ , $a = 3$ , $p = 3$ , therefore

**Vertex**

$(b, a) = (-1, 3)$

**Focus** is

$(b + p, a) = (-1 + 3, 3) = (2, 3)$

And equation of directrix is

$$x = b - p = -1 - (3) \quad \Rightarrow \quad x = -4$$

The given parabola, with its vertex, focus and the directrix is shown below

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