Given a family of lines a(2x +y+4) + b(x-2y-3)=0 .The number of lines belonging to the family at a distance of `sqrt(10)` from point `(2,-3)` is

Options

A. 0
B. 1
C. 2
D. `oo`

Answer

To solve the problem, we need to find the number of lines from the given family of lines that are at a distance of \(\sqrt{10}\) from the point \((2, -3)\). The family of lines is given by the equation:

\[ a(2x + y + 4) + b(x - 2y - 3) = 0 \]

### Step 1: Write the equation of the family of lines
The family of lines can be expressed as:

\[ 2ax + ay + 4a + bx - 2by - 3b = 0 \]

This can be rearranged to:

\[ (2a + b)x + (a - 2b)y + (4a - 3b) = 0 \]

### Step 2: Identify coefficients
From the equation, we can identify the coefficients:
- Coefficient of \(x\): \(A = 2a + b\)
- Coefficient of \(y\): \(B = a - 2b\)
- Constant term: \(C = 4a - 3b\)

### Step 3: Use the distance formula
The distance \(d\) from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is given by:

\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

Substituting the point \((2, -3)\) into the distance formula, we have:

\[ d = \frac{|(2a + b)(2) + (a - 2b)(-3) + (4a - 3b)|}{\sqrt{(2a + b)^2 + (a - 2b)^2}} \]

### Step 4: Simplify the numerator
Calculating the numerator:

\[ = |(2a + b)(2) + (a - 2b)(-3) + (4a - 3b)| \]
\[ = |4a + 2b - 3a + 6b + 4a - 3b| \]
\[ = |(4a - 3a + 4a) + (2b + 6b - 3b)| \]
\[ = |5a + 5b| = 5|a + b| \]

### Step 5: Simplify the denominator
Calculating the denominator:

\[ \sqrt{(2a + b)^2 + (a - 2b)^2} \]
\[ = \sqrt{(4a^2 + 4ab + b^2) + (a^2 - 4ab + 4b^2)} \]
\[ = \sqrt{5a^2 + 5b^2} = \sqrt{5(a^2 + b^2)} \]

### Step 6: Set the distance equal to \(\sqrt{10}\)
Setting the distance equal to \(\sqrt{10}\):

\[ \frac{5|a + b|}{\sqrt{5(a^2 + b^2)}} = \sqrt{10} \]

### Step 7: Cross-multiply and simplify
Cross-multiplying gives:

\[ 5|a + b| = \sqrt{10} \cdot \sqrt{5(a^2 + b^2)} \]
\[ 5|a + b| = \sqrt{50(a^2 + b^2)} \]
\[ 25(a + b)^2 = 50(a^2 + b^2) \]
\[ (a + b)^2 = 2(a^2 + b^2) \]

### Step 8: Rearranging the equation
Rearranging gives:

\[ a^2 + b^2 - 2ab = 0 \]
\[ (a - b)^2 = 0 \]

### Step 9: Conclusion
This implies that \(a = b\). Therefore, there is only one line in the family that is at a distance of \(\sqrt{10}\) from the point \((2, -3)\).

Thus, the number of lines belonging to the family at a distance of \(\sqrt{10}\) from the point \((2, -3)\) is **1**.