Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lie on it

Options

A. `(4sqrt2,2sqrt2)`
B. `(4sqrt3,2sqrt2)`
C. `(4sqrt2,2sqrt3)`
D. `(4sqrt2,2sqrt5)`

Answer

To solve the problem step by step, we will derive the equation of the ellipse based on the given conditions and then check which point lies on it.

### Step 1: Understand the properties of the ellipse
Given that the ellipse has its major axis along the x-axis and is centered at the origin, its standard equation is:

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

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

### Step 2: Use the length of the latus rectum
The length of the latus rectum \(L\) of an ellipse is given by:

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

We know from the problem that the length of the latus rectum is 8. Therefore, we can set up the equation:

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

This simplifies to:

\[ b^2 = 4a \]

### Step 3: Use the relationship between the foci and the minor axis
The distance between the foci of the ellipse is given by \(2c\), where \(c = \sqrt{a^2 - b^2}\). According to the problem, the distance between the foci is equal to the length of the minor axis \(2b\). Therefore, we have:

\[ 2c = 2b \implies c = b \]

Using the relationship \(c = \sqrt{a^2 - b^2}\), we can substitute \(b\) for \(c\):

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

### Step 4: Square both sides
Squaring both sides gives:

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

This simplifies to:

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

### Step 5: Substitute \(b^2\) from earlier
Now we substitute \(b^2 = 4a\) into \(a^2 = 2b^2\):

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

Rearranging gives:

\[ a^2 - 8a = 0 \implies a(a - 8) = 0 \]

Thus, \(a = 0\) (not valid for an ellipse) or \(a = 8\).

### Step 6: Find \(b\)
Now substituting \(a = 8\) back into \(b^2 = 4a\):

\[ b^2 = 4 \times 8 = 32 \implies b = \sqrt{32} = 4\sqrt{2} \]

### Step 7: Write the equation of the ellipse
Now we can write the equation of the ellipse:

\[ \frac{x^2}{8^2} + \frac{y^2}{(4\sqrt{2})^2} = 1 \]

This simplifies to:

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

### Step 8: Check the points
Now we need to check which of the given points lies on the ellipse. Let's check the point \( (4\sqrt{3}, 2\sqrt{2}) \):

Substituting \(x = 4\sqrt{3}\) and \(y = 2\sqrt{2}\):

\[ \frac{(4\sqrt{3})^2}{64} + \frac{(2\sqrt{2})^2}{32} = \frac{48}{64} + \frac{8}{32} = \frac{48}{64} + \frac{8}{32} = \frac{48}{64} + \frac{16}{64} = \frac{64}{64} = 1 \]

Thus, the point \( (4\sqrt{3}, 2\sqrt{2}) \) lies on the ellipse.

### Final Answer:
The point that lies on the ellipse is \( (4\sqrt{3}, 2\sqrt{2}) \).

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