The product of the perpendiculars drawn from the point (1,2) to the pair of lines `x^(2)+4xy+y^(2)=0` is

Options

A. `9//4`
B. `3//4`
C. `9//16`
D. none of these

Explanation

To find the product of the perpendiculars drawn from the point (1, 2) to the pair of lines represented by the equation \(x^2 + 4xy + y^2 = 0\), we can use the formula for the perpendicular distance from a point to a pair of lines.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is \(x^2 + 4xy + y^2 = 0\). Here, we can identify:
- \(a = 1\)
- \(b = 1\)
- \(h = 2\) (since \(2h = 4\))

2. **Identify the point coordinates**:
The point from which the perpendiculars are drawn is \((x_1, y_1) = (1, 2)\).

3. **Use the formula for the product of perpendiculars**:
The product of the perpendiculars from a point \((x_1, y_1)\) to the pair of lines given by \(ax^2 + 2hxy + by^2 = 0\) is given by:
\[ P = \frac{a x_1^2 + 2h x_1 y_1 + b y_1^2}{\sqrt{(a-b)^2 + 4h^2}} \]

4. **Calculate the numerator**:
Substitute \(x_1 = 1\), \(y_1 = 2\), \(a = 1\), \(b = 1\), and \(h = 2\):
\[ P = a x_1^2 + 2h x_1 y_1 + b y_1^2 = 1(1^2) + 2(2)(1)(2) + 1(2^2) \]
\[ = 1 + 8 + 4 = 13 \]

5. **Calculate the denominator**:
\[ \sqrt{(a - b)^2 + 4h^2} = \sqrt{(1 - 1)^2 + 4(2)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]

6. **Calculate the product of the perpendiculars**:
Now substitute the values into the formula:
\[ P = \frac{13}{4} \]

Thus, the product of the perpendiculars drawn from the point (1,2) to the pair of lines \(x^2 + 4xy + y^2 = 0\) is \(\frac{13}{4}\).