If the equation `(10x-5)^(2)+(10y -4)^(2)=lambda^(2) (3x+4y-1)^(2)` represents a hyperbola then

Options

A. `-2 lt lambda lt 2`
B. `lambda gt 2`
C. `lambda gt -2, lambda gt 2`
D. `0 lt lambda lt 2`

Answer

To determine the value of \(\lambda\) for which the equation

\[ (10x - 5)^2 + (10y - 4)^2 = \lambda^2 (3x + 4y - 1)^2 \]

represents a hyperbola, we can follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten as:

\[ (10x - 5)^2 + (10y - 4)^2 = \lambda^2 (3x + 4y - 1)^2 \]

### Step 2: Identify the standard form
To analyze the equation, we can compare it to the standard form of a conic section. The left side represents a circle (or ellipse) and the right side represents a squared term. For the equation to represent a hyperbola, the term on the left must be greater than the term on the right.

### Step 3: Factor out constants
We can factor out constants from both sides to simplify the equation.

From the left side:

\[ (10x - 5)^2 + (10y - 4)^2 = 100\left(\left(x - \frac{1}{2}\right)^2 + \left(y - \frac{2}{5}\right)^2\right) \]

From the right side, we can express it as:

\[ \lambda^2 (3x + 4y - 1)^2 \]

### Step 4: Set up the relationship for hyperbola
For the equation to represent a hyperbola, we need to ensure that the eccentricity \(e\) is greater than 1. The eccentricity for a hyperbola can be expressed as:

\[ e = \frac{\text{Distance from center to focus}}{\text{Distance from center to vertex}} \]

From the standard form, we can relate it to:

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

### Step 5: Establish the inequality
From our earlier work, we have:

\[ 100\left(\left(x - \frac{1}{2}\right)^2 + \left(y - \frac{2}{5}\right)^2\right) > \lambda^2 (3x + 4y - 1)^2 \]

To ensure this represents a hyperbola, we need:

\[ \frac{\lambda^2}{100} < 1 \]

### Step 6: Solve for \(\lambda\)
This leads us to:

\[ \lambda^2 < 100 \implies |\lambda| < 10 \]

### Step 7: Determine the conditions for \(\lambda\)
Since we need the eccentricity to be greater than 1, we also have:

\[ |\lambda| > 2 \]

### Conclusion
Combining both conditions, we find:

\[ \lambda < -2 \quad \text{or} \quad \lambda > 2 \]

Thus, the values of \(\lambda\) for which the equation represents a hyperbola are:

\[ \lambda < -2 \quad \text{or} \quad \lambda > 2 \]