Question

A person standing at the crossing at two straight paths represented by the equations 2x-3y-4 = 0 and 3x-4y-5 = 0, wants to reach a path represented by 6x-7y+8 = 0 in least time. Find the equations of path he should follow.

Answer 1

Answer on Question #69436 – Math – Other

Question

A person standing at the crossing at two straight paths represented by the equations $2x-3y-4 = 0$ and $3x-4y-5 = 0$, wants to reach a path represented by $6x-7y+8 = 0$ in least time. Find the equations of path he should follow.

Solution

First, let's find where the person is standing. This is an intersection of two lines, which can be found by solving the system of equations representing these lines:

$$\begin{array}{l} \left\{ \begin{array}{l} 2x - 3y - 4 = 0 \\ 3x - 4y - 5 = 0 \end{array} \right. \rightarrow \left\{ \begin{array}{l} x = \frac{1}{2}(3y + 4) \\ 3x - 4y - 5 = 0 \end{array} \right. \rightarrow \left\{ \begin{array}{l} x = \frac{1}{2}(3y + 4) \\ \frac{3}{2}(3y + 4) - 4y - 5 = 0 \end{array} \right. \rightarrow \left\{ \begin{array}{l} x = \frac{1}{2}(3y + 4) \\ \frac{9}{2}y + 6 - 4y - 5 = 0 \end{array} \right. \\ \rightarrow \left\{ \begin{array}{l} x = \frac{1}{2}(3y + 4) \\ \frac{1}{2}y + 1 = 0 \end{array} \right. \rightarrow \left\{ \begin{array}{l} x = \frac{1}{2}(3y + 4) \\ y = -2 \end{array} \right. \rightarrow \left\{ \begin{array}{l} x = \frac{1}{2}(3(-2) + 4) \\ y = -2 \end{array} \right. \rightarrow \left\{ \begin{array}{l} x = -1 \\ y = -2 \end{array} \right. \end{array}$$

The person is initially standing at $(x_0, y_0) = (-1, -2)$.

The shortest path to new $3^{\text{rd}}$ path is perpendicular to it. For a line represented in form $ax + by + c = 0$ the perpendicular must follow the following form $-by + ax + d = 0$. $3^{\text{rd}}$ path is represented by $6x - 7y + 8 = 0$, so perpendicular to it is $7x + 6y + d = 0$ for some unknown $d$. To find $d$ let's substitute initial place into equation – this path must contain it:

$$7x_0 + 6y_0 + d = 0 \rightarrow 7(-1) + 6(-2) + d = 0 \rightarrow -19 + d = 0 \rightarrow d = 19.$$

The person must follow the path represented by $7x + 6y + 19 = 0$.

**Answer**: $7x + 6y + 19 = 0$.

Answer provided by https://www.AssignmentExpert.com