If a line `(x)/(8)+(y)/(5)=1` meets the ellipse on x - axis and the line `(x)/(3)+(y)/(4) =1` meets the ellipse on y - axis then its eccentricity

Options

A. `(1)/sqrt(2)`
B. `sqrt(2/3)`
C. `sqrt(3/4)`
D. `sqrt(4/5)`

Answer

To solve the problem step by step, we will follow the given information and derive the eccentricity of the ellipse.

### Step 1: Identify the points where the lines meet the axes
The first line given is:
\[ \frac{x}{8} + \frac{y}{5} = 1 \]
To find where this line meets the x-axis, we set \(y = 0\):
\[ \frac{x}{8} + \frac{0}{5} = 1 \implies \frac{x}{8} = 1 \implies x = 8 \]
So, the point where this line meets the x-axis is \( (8, 0) \).

### Step 2: Identify the point where the second line meets the y-axis
The second line given is:
\[ \frac{x}{3} + \frac{y}{4} = 1 \]
To find where this line meets the y-axis, we set \(x = 0\):
\[ \frac{0}{3} + \frac{y}{4} = 1 \implies \frac{y}{4} = 1 \implies y = 4 \]
So, the point where this line meets the y-axis is \( (0, 4) \).

### Step 3: Determine the semi-major and semi-minor axes
From the points found:
- The point \( (8, 0) \) indicates that the semi-major axis \( A = 8 \).
- The point \( (0, 4) \) indicates that the semi-minor axis \( B = 4 \).

### Step 4: Calculate the eccentricity of the ellipse
The formula for the eccentricity \( e \) of an ellipse is given by:
\[ e = \sqrt{1 - \frac{B^2}{A^2}} \]
Substituting the values of \( A \) and \( B \):
\[ e = \sqrt{1 - \frac{4^2}{8^2}} = \sqrt{1 - \frac{16}{64}} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Final Answer
The eccentricity of the ellipse is:
\[ e = \frac{\sqrt{3}}{2} \]

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