For the ellipse ` (x^(2))/(4) + (y^(2))/(3) = 1` , the ends of the two latus rectum are the four points

Options

A. `+- 1,+-3/2`
B. `+-3/2,+- 1`
C. `+-2/3,+- 1`
D. `+- 1,+-2/3`

Answer

To find the ends of the two latus rectum of the ellipse given by the equation

\[ \frac{x^2}{4} + \frac{y^2}{3} = 1, \]

we can follow these steps:

### Step 1: Identify the values of \(a\) and \(b\)

The standard form of the ellipse is

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

From the given equation, we can identify:

- \(a^2 = 4 \Rightarrow a = 2\)
- \(b^2 = 3 \Rightarrow b = \sqrt{3}\)

### Step 2: Calculate the eccentricity \(e\)

The eccentricity \(e\) of the ellipse is given by the formula:

\[ e = \sqrt{1 - \frac{b^2}{a^2}}. \]

Substituting the values of \(b^2\) and \(a^2\):

\[ e = \sqrt{1 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2}. \]

### Step 3: Find the coordinates of the foci

The coordinates of the foci of the ellipse are given by:

\[ (\pm ae, 0). \]

Substituting the values of \(a\) and \(e\):

\[ \text{Foci} = (\pm 2 \cdot \frac{1}{2}, 0) = (\pm 1, 0). \]

### Step 4: Determine the ends of the latus rectum

The ends of the latus rectum can be found using the formula:

\[ \left( \pm ae, \pm \frac{b^2}{a} \right). \]

Calculating \(b^2/a\):

\[ \frac{b^2}{a} = \frac{3}{2}. \]

Now substituting the values:

1. For the right focus \((ae, 0) = (1, 0)\):
- Ends of the latus rectum: \((1, \frac{3}{2})\) and \((1, -\frac{3}{2})\).

2. For the left focus \((-ae, 0) = (-1, 0)\):
- Ends of the latus rectum: \((-1, \frac{3}{2})\) and \((-1, -\frac{3}{2})\).

### Step 5: Compile the final points

Thus, the four points that are the ends of the latus rectum are:

\[ (1, \frac{3}{2}), (1, -\frac{3}{2}), (-1, \frac{3}{2}), (-1, -\frac{3}{2}). \]

### Summary of the solution

The ends of the two latus rectum of the ellipse \(\frac{x^2}{4} + \frac{y^2}{3} = 1\) are:

1. \((1, \frac{3}{2})\)
2. \((1, -\frac{3}{2})\)
3. \((-1, \frac{3}{2})\)
4. \((-1, -\frac{3}{2})\)