Question:

The shortest distance between the lines $\frac{x-5}{4}=\frac{y-7}{-5}=\frac{z+3}{-5}$ and $\frac{x-8}{7}=\frac{y-7}{1}=\frac{z-5}{-3}$

Updated On: Jul 7, 2022

$\frac{372\sqrt{2}}{35}$
$\frac{186}{35}$
$\frac{372}{35\sqrt{2}}$
$\frac{186}{35\sqrt{2}}$

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The Correct Option is C

Solution and Explanation

Here, $x_1 = 5$, $y_1 = 7$, $z_1 = -3$, $a_1 = 4$, $b_1 = -5$, $c_1= -5$ and $x_2= 8$, $y_2 = 7$, $z_2 = 5$, $a_2 = 7$, $b_2 = 1$, $c_2 = -3$ $d=\left|\frac{\begin{vmatrix}3&0&8\\ 4&-5&-5\\ 7&1&-3\end{vmatrix}}{\sqrt{\left(15+5\right)^{2}+\left(-35+12\right)^{2}+\left(4+35\right)^{2}}}\right|$ $=\left|\frac{3\left(15+5\right)+8\left(4+35\right)}{\sqrt{400+529+1521}}\right|$ $=\left|\frac{60+312}{\sqrt{2450}}\right|$ $=\frac{372}{35\sqrt{2}}$

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Notes on Straight lines

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