Find the coordinates of a point on x + y + 3 = 0, whose distance from x + 2y + 2 = 0 is `sqrt(5)` .

Answer

To find the coordinates of a point on the line \( x + y + 3 = 0 \) whose distance from the line \( x + 2y + 2 = 0 \) is \( \sqrt{5} \), we can follow these steps:

### Step 1: Define the point on the first line
Let the required point be \( (a, b) \). Since this point lies on the line \( x + y + 3 = 0 \), we can express this relationship as:
\[ a + b + 3 = 0 \quad \text{(Equation 1)} \]
From this, we can derive:
\[ a + b = -3 \quad \text{(1)} \]

### Step 2: Use the distance formula from a point to a line
The distance \( d \) from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \) is given by the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( x + 2y + 2 = 0 \), we have \( A = 1 \), \( B = 2 \), and \( C = 2 \). The distance from the point \( (a, b) \) to this line is given to be \( \sqrt{5} \):
\[ \frac{|1 \cdot a + 2 \cdot b + 2|}{\sqrt{1^2 + 2^2}} = \sqrt{5} \]
This simplifies to:
\[ \frac{|a + 2b + 2|}{\sqrt{5}} = \sqrt{5} \]

### Step 3: Simplify the distance equation
Multiplying both sides by \( \sqrt{5} \) gives:
\[ |a + 2b + 2| = 5 \]
This leads to two cases:
1. \( a + 2b + 2 = 5 \)
2. \( a + 2b + 2 = -5 \)

### Step 4: Solve the first case
**Case 1:**
\[ a + 2b + 2 = 5 \]
Subtracting 2 from both sides:
\[ a + 2b = 3 \quad \text{(Equation 2)} \]

Now we have two equations:
1. \( a + b = -3 \) (from Equation 1)
2. \( a + 2b = 3 \) (from Equation 2)

We can solve these equations simultaneously. Subtract Equation 1 from Equation 2:
\[ (a + 2b) - (a + b) = 3 - (-3) \]
This simplifies to:
\[ b = 6 \]
Substituting \( b = 6 \) back into Equation 1:
\[ a + 6 = -3 \implies a = -9 \]
Thus, one point is \( (-9, 6) \).

### Step 5: Solve the second case
**Case 2:**
\[ a + 2b + 2 = -5 \]
Subtracting 2 from both sides:
\[ a + 2b = -7 \quad \text{(Equation 3)} \]

Now we have:
1. \( a + b = -3 \) (from Equation 1)
2. \( a + 2b = -7 \) (from Equation 3)

Subtract Equation 1 from Equation 3:
\[ (a + 2b) - (a + b) = -7 - (-3) \]
This simplifies to:
\[ b = -4 \]
Substituting \( b = -4 \) back into Equation 1:
\[ a - 4 = -3 \implies a = 1 \]
Thus, the second point is \( (1, -4) \).

### Final Answer
The required points are:
\[ (-9, 6) \quad \text{and} \quad (1, -4) \]