Let the length of latus rectum of an ellipse with its major axis along x-axis and center at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of the minor axis , then which of the following points lies on it: (a) `(4sqrt2, 2sqrt2)` (b) `(4sqrt3, 2sqrt2)` (c) `(4sqrt3, 2sqrt3)` (d) `(4sqrt2, 2sqrt3)`

Options

A. `(4 sqrt(3), 2sqrt(3))`
B. `(4sart(3),2sqrt(2))`
C. `(4sqrt(2), 2sqrt(2))`
D. `(4sqrt(2), 2sqrt(3))`

Answer

To solve the given problem step by step, we will follow the mathematical reasoning and calculations as outlined in the video transcript.

### Step 1: Understand the properties of the ellipse

The standard form of an ellipse with its major axis along the x-axis is given by:

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

where \(2a\) is the length of the major axis and \(2b\) is the length of the minor axis.

### Step 2: Use the information about the latus rectum

The length of the latus rectum \(L\) of an ellipse is given by:

\[ L = \frac{2b^2}{a} \]

According to the problem, the length of the latus rectum is 8, so we have:

\[ \frac{2b^2}{a} = 8 \]

This simplifies to:

\[ b^2 = 4a \]

### Step 3: Use the information about the distance between the foci

The distance between the foci \(2c\) of the ellipse is given by:

\[ c = \sqrt{a^2 - b^2} \]

The problem states that the distance between the foci is equal to the length of the minor axis \(2b\), so we have:

\[ 2c = 2b \]

This simplifies to:

\[ c = b \]

### Step 4: Relate \(c\), \(a\), and \(b\)

From the definition of \(c\):

\[ c = \sqrt{a^2 - b^2} \]

Substituting \(c = b\) into the equation gives:

\[ b = \sqrt{a^2 - b^2} \]

Squaring both sides:

\[ b^2 = a^2 - b^2 \]

This leads to:

\[ 2b^2 = a^2 \quad \text{or} \quad a^2 = 2b^2 \]

### Step 5: Substitute \(b^2\) from earlier

We already found that \(b^2 = 4a\). Now substituting this into \(a^2 = 2b^2\):

\[ a^2 = 2(4a) = 8a \]

Rearranging gives:

\[ a^2 - 8a = 0 \]

Factoring out \(a\):

\[ a(a - 8) = 0 \]

Thus, \(a = 0\) or \(a = 8\). Since \(a = 0\) is not valid for an ellipse, we have:

\[ a = 8 \]

### Step 6: Find \(b^2\)

Using \(b^2 = 4a\):

\[ b^2 = 4(8) = 32 \]

### Step 7: Write the equation of the ellipse

Now we can write the equation of the ellipse:

\[ \frac{x^2}{64} + \frac{y^2}{32} = 1 \]

### Step 8: Check which points lie on the ellipse

We will check each of the given points to see if they satisfy the ellipse equation.

1. **Point (4√2, 2√2)**:
\[ \frac{(4\sqrt{2})^2}{64} + \frac{(2\sqrt{2})^2}{32} = \frac{32}{64} + \frac{8}{32} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \quad \text{(not on the ellipse)} \]

2. **Point (4√3, 2√2)**:
\[ \frac{(4\sqrt{3})^2}{64} + \frac{(2\sqrt{2})^2}{32} = \frac{48}{64} + \frac{8}{32} = \frac{3}{4} + \frac{1}{4} = 1 \quad \text{(on the ellipse)} \]

3. **Point (4√3, 2√3)**:
\[ \frac{(4\sqrt{3})^2}{64} + \frac{(2\sqrt{3})^2}{32} = \frac{48}{64} + \frac{12}{32} = \frac{3}{4} + \frac{3}{8} = \frac{3}{4} + \frac{9}{24} = \frac{3}{4} + \frac{3}{8} \quad \text{(not on the ellipse)} \]

4. **Point (4√2, 2√3)**:
\[ \frac{(4\sqrt{2})^2}{64} + \frac{(2\sqrt{3})^2}{32} = \frac{32}{64} + \frac{12}{32} = \frac{1}{2} + \frac{3}{8} = \frac{4}{8} + \frac{3}{8} = \frac{7}{8} \quad \text{(not on the ellipse)} \]

### Conclusion

The point that lies on the ellipse is **(4√3, 2√2)**.