Question

The least distance from origin to a point on the line \(y=x+3\) which lies at a distance of \(2\) units from \((0,3)\) is

• A. \(13+6\sqrt{2}\)
• B. \(10+6\sqrt{2}\)
• C. \(10-6\sqrt{2}\)
• D. \(13-6\sqrt{2}\)

Hint:

When comparing distances, it is often easier to compare the squares of the distances instead of the actual distances.

Answer 1

The Correct Option is D

Step 1: Let the required point on the line be determined.
Given line is \[ y=x+3 \] Any point on this line can be written as \[ (x,x+3) \] The point is at a distance \(2\) units from \((0,3)\).
Using the distance formula, \[ \sqrt{(x-0)^2+\big((x+3)-3\big)^2}=2 \] \[ \sqrt{x^2+x^2}=2 \] \[ \sqrt{2x^2}=2 \] Squaring both sides, \[ 2x^2=4 \] \[ x^2=2 \] \[ x=\pm \sqrt{2} \] Hence, the two possible points are \[ (\sqrt{2},\,3+\sqrt{2}) \] and \[ (-\sqrt{2},\,3-\sqrt{2}) \]

Step 2: Find the distance of these points from the origin.
Distance squared from origin to the point \[ (\sqrt{2},\,3+\sqrt{2}) \] is \[ (\sqrt{2})^2+(3+\sqrt{2})^2 \] \[ =2+9+2+6\sqrt{2} \] \[ =13+6\sqrt{2} \] Distance squared from origin to the point \[ (-\sqrt{2},\,3-\sqrt{2}) \] is \[ (-\sqrt{2})^2+(3-\sqrt{2})^2 \] \[ =2+9+2-6\sqrt{2} \] \[ =13-6\sqrt{2} \]

Step 3: Determine the least distance.
Since \[ 13-6\sqrt{2}\lt 13+6\sqrt{2}, \] the least distance corresponds to \[ 13-6\sqrt{2} \]

Step 4: Final conclusion.
Therefore, the required answer is \[ \boxed{13-6\sqrt{2}} \]