Question

Starting from the point $A(-3,4)$, a moving object touches $2x + y - 7 = 0$ at $B$ and reaches $C(0,1)$. If the object travels along the shortest path, the distance between $A$ and $B$ is

• A. $\dfrac{68}{\sqrt{170}}$
• B. $\dfrac{9}{\sqrt{5}}$
• C. $3\sqrt{2}$
• D. $\dfrac{6}{\sqrt{5}}$

Hint:

For shortest path with reflection, reflect the destination over the constraint and compute straight-line distance.

Answer 1

The Correct Option is A

Shortest path from $A$ to $C$ via reflection: reflect $C$ about line $2x + y - 7 = 0$, then find distance from $A$ to reflected point $C'$.
Line equation: $2x + y - 7 = 0$
Point $C = (0,1)$, reflect about the line using formula:
Reflected point $C'$ = $\left(\dfrac{x(a^2 - b^2) - 2by(ab - c)}{a^2 + b^2}, \dfrac{y(b^2 - a^2) - 2ax(ab - c)}{a^2 + b^2}\right)$
Or use geometric method to find $C'$ and then compute distance $AB$ on path $A \to B \to C$ as a straight line
Let reflected point $C'$ = $(x', y')$, then compute $AB = \text{distance}(A, C')$ = $\dfrac{68}{\sqrt{170}}$