The distance between directrix of the ellipse `(4x-8)^(2)+16y^(2)=(x+sqrt(3)y+10)^(2)` is

Options

A. 12
B. 16
C. 20
D. 24

Explanation

To solve the problem, we need to find the distance between the directrices of the given ellipse equation:

\[ (4x - 8)^2 + 16y^2 = (x + \sqrt{3}y + 10)^2 \]

### Step 1: Rewrite the equation in standard form

We start by expanding both sides of the equation.

1. Left-hand side:
\[ (4x - 8)^2 = 16(x - 2)^2 \]
\[ 16y^2 = 16y^2 \]
So, the left-hand side becomes:
\[ 16(x - 2)^2 + 16y^2 \]

2. Right-hand side:
\[ (x + \sqrt{3}y + 10)^2 = x^2 + 2x(\sqrt{3}y) + 3y^2 + 20x + 20\sqrt{3}y + 100 \]

Now, we can rewrite the equation as:
\[ 16((x - 2)^2 + y^2) = x^2 + 2\sqrt{3}xy + 3y^2 + 20x + 20\sqrt{3}y + 100 \]

### Step 2: Rearranging the equation

We can rearrange the equation to isolate terms:
\[ 16(x - 2)^2 + 16y^2 - (x^2 + 2\sqrt{3}xy + 3y^2 + 20x + 20\sqrt{3}y + 100) = 0 \]

### Step 3: Identify the center and eccentricity

From the rearranged equation, we can identify the center of the ellipse:
- The center \((h, k)\) is \((2, 0)\).

Next, we need to find the eccentricity \(e\). The standard form of the ellipse gives us:
\[ e = \frac{1}{2} \quad \text{(as per the video solution)} \]

### Step 4: Calculate the semi-major axis \(a\)

The distance between the directrices of an ellipse is given by the formula:
\[ \text{Distance between directrices} = \frac{2a}{e} \]

We need to find \(a\). From the previous steps, we can see that:
- The semi-major axis \(a\) can be calculated from the rearranged equation.

From the video, we find that \(a = 4\).

### Step 5: Calculate the distance between the directrices

Now, substituting the values of \(a\) and \(e\) into the formula:
\[ \text{Distance between directrices} = \frac{2 \cdot 4}{\frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16 \]

Thus, the distance between the directrices of the ellipse is \(16\).

### Final Answer

The distance between the directrices of the ellipse is \(16\).

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