The value of `lamda ` for which the equation `(10 x -5) ^(2) + (10y-15) ^(2) =lamda^(2)(5x + 12 y +7) ^(2)` represents a parabola is

Options

A. `10/13`
B. `-10/13`
C. `13/10`
D. none of these

Answer

To find the value of `λ` for which the equation

\[ (10x - 5)^2 + (10y - 15)^2 = λ^2(5x + 12y + 7)^2 \]

represents a parabola, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:

\[ (10x - 5)^2 + (10y - 15)^2 = λ^2(5x + 12y + 7)^2 \]

### Step 2: Expand both sides
Expanding the left-hand side:

\[ (10x - 5)^2 = 100x^2 - 100x + 25 \]
\[ (10y - 15)^2 = 100y^2 - 300y + 225 \]

Combining these:

\[ 100x^2 + 100y^2 - 100x - 300y + 250 \]

Now the left-hand side becomes:

\[ 100(x^2 + y^2) - 100x - 300y + 250 \]

Expanding the right-hand side:

\[ λ^2(5x + 12y + 7)^2 = λ^2(25x^2 + 120xy + 144y^2 + 70x + 168y + 49) \]

### Step 3: Set up the condition for a parabola
For the equation to represent a parabola, the discriminant of the resulting quadratic equation must equal zero. The general form of a conic section is given by:

\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

The condition for a parabola is:

\[ B^2 - 4AC = 0 \]

### Step 4: Identify coefficients
From the expanded forms, we can identify coefficients \(A\), \(B\), and \(C\) from both sides.

From the left side:
- \(A = 100\)
- \(B = 0\)
- \(C = 100\)

From the right side, we need to collect terms and identify \(A\), \(B\), and \(C\) in terms of \(λ\).

### Step 5: Solve for λ
Setting the discriminant condition \(B^2 - 4AC = 0\) gives us:

\[ 0^2 - 4(100)(\text{coefficient of } y^2) = 0 \]

This will lead us to find the value of \(λ\).

### Step 6: Calculate λ
After simplifying and solving the equation, we find:

\[ λ^2 = \frac{100}{169} \]

Taking the square root gives:

\[ λ = \pm \frac{10}{13} \]

Since we are looking for a positive value (as eccentricity must be positive), we take:

\[ λ = \frac{10}{13} \]

### Final Answer
Thus, the value of `λ` for which the equation represents a parabola is:

\[ λ = \frac{10}{13} \]

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