A person standing at the junction (crossing) of two straight paths represented by the equations 2x - 3y + 4 = 0 and 3x + 4y - 5 = 0 wants to reach the path whose equation is 6x - 7y + 8 = 0 in the least time. Find equation of the path that he should follow.
Solution and Explanation
The equations of the given lines are
\(2x - 3y + 4 = 0 … (1)\)
\(3x + 4y - 5 = 0 … (2)\)
\(6x - 7y + 8 = 0 … (3)\)
The person is standing at the junction of the paths represented by lines (1) and (2). On solving equations (1) and (2), we obtain \(x = -\frac{ 1}{17}\) and \(y=\frac{22}{7}\)
Thus, the person is standing at point \((\frac{-1}{17}, \frac{22}{17})\).
The person can reach path (3) in the least time if he walks along the perpendicular line to (3) from point \((\frac{-1}{17}, \frac{22}{17})\)
Slope of the line \((3)=\frac{6}{7}\)
∴Slope of the line perpendicular to line (3) \(=\frac{-1}{(\frac{6}{7})}=– \frac{7}{6}\)
The equation of the line passing through \((\frac{-1}{17}, \frac{22}{17})\) and having a slope of \(\frac{-7}{6}\) is given by
\((y-\frac{22}{17}) =\frac{-7}{6}(x+\frac{1}{17})\)
\(6 (17y – 22) = – 7 (17x + 1)\)
\(102y – 132 = – 119x – 7\)
\(1119x + 102y = 125\)
Hence, the path that the person should follow is \(119x + 102y = 125.\)
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Concepts Used:
Straight lines
A straight line is a line having the shortest distance between two points.
A straight line can be represented as an equation in various forms, as show in the image below:
The following are the many forms of the equation of the line that are presented in straight line-
1. Slope – Point Form
Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.
y – y0 = m (x – x0)
2. Two – Point Form
Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2) are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes
The slope of P2P = The slope of P1P2 , i.e.
\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)
Hence, the equation becomes:
y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)
3. Slope-Intercept Form
Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by
y – c =m( x - 0 )
As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if
y = m x +c