Latusrectum of the conic satisfying the differential equation `xdy+ydx=0` and passing through the point (2, 8) is

Options

A. `4sqrt(2)`
B. `8`
C. `8sqrt(2)`
D. `16`

Answer

To solve the problem of finding the length of the latus rectum of the conic that satisfies the differential equation \( x dy + y dx = 0 \) and passes through the point (2, 8), we can follow these steps:

### Step 1: Rewrite the Differential Equation
The given differential equation is:
\[ x dy + y dx = 0 \]
We can rearrange this equation by dividing both sides by \( xy \):
\[ \frac{dy}{y} + \frac{dx}{x} = 0 \]

**Hint:** To manipulate the equation, consider separating the variables.

### Step 2: Integrate Both Sides
Now, we will integrate both sides:
\[ \int \frac{dy}{y} + \int \frac{dx}{x} = 0 \]
This results in:
\[ \ln |y| + \ln |x| = \ln |c| \]
where \( c \) is the constant of integration.

**Hint:** Remember that the integral of \( \frac{1}{u} \) is \( \ln |u| \).

### Step 3: Simplify the Equation
Using properties of logarithms, we can combine the logarithmic terms:
\[ \ln |xy| = \ln |c| \]
Exponentiating both sides gives us:
\[ xy = c \]

**Hint:** Exponentiation helps eliminate the logarithm, leading to a product form.

### Step 4: Identify the Conic
The equation \( xy = c \) represents a rectangular hyperbola.

**Hint:** Recognize that this is a standard form of a hyperbola.

### Step 5: Find the Constant \( c \)
Since the hyperbola passes through the point (2, 8), we can substitute these values into the equation:
\[ 2 \cdot 8 = c \implies c = 16 \]

**Hint:** Substitute the coordinates of the given point directly into the equation.

### Step 6: Write the Equation of the Hyperbola
Now we can write the equation of the hyperbola:
\[ xy = 16 \]

**Hint:** Use the value of \( c \) to express the hyperbola.

### Step 7: Find the Length of the Latus Rectum
For a rectangular hyperbola of the form \( xy = c^2 \), the length of the latus rectum is given by:
\[ \text{Length of Latus Rectum} = 2\sqrt{2}c \]
Substituting \( c = 4 \) (since \( c^2 = 16 \)):
\[ \text{Length of Latus Rectum} = 2\sqrt{2} \cdot 4 = 8\sqrt{2} \]

**Hint:** Remember to use the correct formula for the latus rectum based on the hyperbola's equation.

### Final Answer
The length of the latus rectum of the conic is:
\[ \boxed{8\sqrt{2}} \]