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:

Translate geometric conditions into algebra and minimize the distance using coordinates.

Answer 1

The Correct Option is D

Let point $P$ be on line $y = x + 3$, and $P$ lies 2 units from $(0,3)$.
Let $P = (x, x+3)$. Then distance from $(0,3)$ is: $\sqrt{x^2 + (x+3 - 3)^2} = \sqrt{x^2 + x^2} = \sqrt{2}x$
Given $\sqrt{2}x = 2 \Rightarrow x = \dfrac{2}{\sqrt{2}} = \sqrt{2}$
So $P = (\sqrt{2}, \sqrt{2} + 3)$
Distance from origin to $P$ = $\sqrt{(\sqrt{2})^2 + (\sqrt{2}+3)^2} = \sqrt{2 + (3+\sqrt{2})^2}$
$= \sqrt{2 + 9 + 6\sqrt{2} + 2} = \sqrt{13 + 6\sqrt{2}}$
Among the options, the least distance is $13 - 6\sqrt{2}$