Consider the following points :
I (0,5)
II (2 , -1)
III (3 , -4)
Which of the above lie on the line 3x + y = 5 and at distance `sqrt(10)` from (1 , 2) ?

Options

A. Only I
B. Only II
C. I and II
D. I , II and III

Answer

To solve the problem, we need to determine which of the given points lies on the line defined by the equation \(3x + y = 5\) and is at a distance of \(\sqrt{10}\) from the point \((1, 2)\).

### Step 1: Check if each point lies on the line \(3x + y = 5\)

**Point I: (0, 5)**
- Substitute \(x = 0\) and \(y = 5\) into the equation:
\[ 3(0) + 5 = 5 \]
- LHS = 5, RHS = 5. This point lies on the line.

**Point II: (2, -1)**
- Substitute \(x = 2\) and \(y = -1\) into the equation:
\[ 3(2) + (-1) = 6 - 1 = 5 \]
- LHS = 5, RHS = 5. This point lies on the line.

**Point III: (3, -4)**
- Substitute \(x = 3\) and \(y = -4\) into the equation:
\[ 3(3) + (-4) = 9 - 4 = 5 \]
- LHS = 5, RHS = 5. This point lies on the line.

### Step 2: Calculate the distance from each point to (1, 2)

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

**Distance from Point I (0, 5) to (1, 2)**
- Using the distance formula:
\[ D = \sqrt{(0 - 1)^2 + (5 - 2)^2} = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
- This distance is \(\sqrt{10}\).

**Distance from Point II (2, -1) to (1, 2)**
- Using the distance formula:
\[ D = \sqrt{(2 - 1)^2 + (-1 - 2)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
- This distance is \(\sqrt{10}\).

**Distance from Point III (3, -4) to (1, 2)**
- Using the distance formula:
\[ D = \sqrt{(3 - 1)^2 + (-4 - 2)^2} = \sqrt{(2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \]
- This distance is not \(\sqrt{10}\).

### Conclusion
The points that lie on the line \(3x + y = 5\) and are at a distance of \(\sqrt{10}\) from the point \((1, 2)\) are:
- Point I: (0, 5)
- Point II: (2, -1)