Question

The distance between the point \( (1,2) \) and the point of intersection of the lines \( 2x + y = 2 \) and \( x + 2y = 2 \) is

• A. \( \frac{\sqrt{17}}{3} \)
• B. \( \frac{\sqrt{16}}{3} \)
• C. \( \frac{\sqrt{17}}{5} \)
• D. \( \frac{\sqrt{19}}{3} \)
• E. \( \frac{\sqrt{19}}{5} \)

Hint:

Always solve simultaneous equations carefully—small arithmetic mistakes lead to wrong distance.

Answer 1

The Correct Option is A

Concept:
• To find intersection point, solve simultaneous equations.
• Distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 1: Solving the given equations.
Given: \[ 2x + y = 2 \quad ...(1) \] \[ x + 2y = 2 \quad ...(2) \] Multiply equation (2) by 2: \[ 2x + 4y = 4 \quad ...(3) \]

Step 2: Subtracting equations.
Subtract (1) from (3): \[ (2x + 4y) - (2x + y) = 4 - 2 \] \[ 3y = 2 \Rightarrow y = \frac{2}{3} \]

Step 3: Finding \(x\).
Substitute into (1): \[ 2x + \frac{2}{3} = 2 \] \[ 2x = \frac{6 - 2}{3} = \frac{4}{3} \] \[ x = \frac{2}{3} \] Thus, intersection point: \[ \left( \frac{2}{3}, \frac{2}{3} \right) \]

Step 4: Applying distance formula.
Distance from \( (1,2) \): \[ d = \sqrt{\left(1 - \frac{2}{3}\right)^2 + \left(2 - \frac{2}{3}\right)^2} \]

Step 5: Simplifying.
\[ = \sqrt{\left(\frac{1}{3}\right)^2 + \left(\frac{4}{3}\right)^2} \] \[ = \sqrt{\frac{1}{9} + \frac{16}{9}} = \sqrt{\frac{17}{9}} \] \[ = \frac{\sqrt{17}}{3} \]

Step 6: Final Answer.
\[ \boxed{\frac{\sqrt{17}}{3}} \]