Question

The shortest distance between the z-axis and the line \[x + y + 2z - 3 = 0,2x + 3y + 4z - 4 = 0\] is
 A. 1
B. 2
C. 4
D. 3

Answer 1

Hint:In the general equation of a line, \[ax\; + \;by\; + \;c\; = 0\] , a and b cannot both be zero unless c is also zero, in which case the equation does not define a line. If a = 0 and b ≠ 0, the line is horizontal and has equation \[y\; = - c/b\]. The distance from (x0, y0) to this line is measured along a vertical line segment of length \[\left| {{y_0}\; - {\text{ }}\dfrac{{( - c)}}{b}} \right|{\text{ = }}\dfrac{{\left| {b{y_0}\; + \;c} \right|}}{{|b|}}\] in accordance with the formula. Similarly, for vertical lines (b = 0), the distance formula between the same point and the line is \[\dfrac{{\left| {a{x_0}\; + \;c} \right|}}{{|a|}}\] , as measured along the horizontal line segment.

Complete step-by-step solution
Direction ratios of the line segments \[x + y + 2z - 3 = 0,2x + 3y + 4z - 4 = 0\]is given as: $D{R_1} = (1,1,2){\text{ and, }}D{R_2} = (2,3,4)$.
The direction ratio of the line passing through the intersection of these two lines is given by the cross-product of the direction ratio of the given lines.
$
  DR = \left| {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  1&1&2 \\
  2&3&4
\end{array}} \right| \\
   = \hat i\left( {4 - 6} \right) + \hat j\left( {4 - 4} \right) + \hat k\left( {3 - 2} \right) \\
   = - 2\hat i + \hat k \\
 $
Hence, the direction ration can be written as $\left( {m,n,l} \right) = \left( { - 2,0,1} \right) - - - - (i)$
As the line is perpendicular to the z-axis so, the value of $z = 0$ will satisfy both the given equations as:
$x + y = 3$ and $2x + 3y = 4$
Solving the above two equations by substituting $x = - y$ in the equation $2x + 3y = 4$as:
$
  2(3 - y) + 3y = 4 \\
  6 - 2y + 3y = 4 \\
  6 + y = 4 \\
  y = - 2 \\
 $
Substituting the value of $y = - 2$ in the equation $x + y = 3$ to determine the value of $x$ as:
$
  x = 3 - y \\
   = 3 - ( - 2) \\
   = 5 \\
 $
Hence, the points through which the required line is passing is $\left( {{x_1},{y_1},{z_1}} \right) \equiv \left( {5, - 2,0} \right)$ and its direction ratio is $\left( {m,n,l} \right) \equiv \left( { - 2,0,1} \right)$
The equation of the line is given as: $\dfrac{{x - {x_1}}}{m} = \dfrac{{y - {y_1}}}{n} = \dfrac{{z - {z_1}}}{l}$
Thus its equation may be written as, \[\dfrac{{x - 5}}{{ - 2}} = \dfrac{{y + 2}}{0} = \dfrac{{z - 0}}{1} - - - - (ii)\]
and equation of line z-axis is, \[\dfrac{x}{0} = \dfrac{y}{0} = \dfrac{z}{1} - - - - (iii)\]
Now we know the shortest distance between two lines is given by, $D = \dfrac{{\left| {\begin{array}{*{20}{c}}
  {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\
  {{a_1}}&{{b_1}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}}
\end{array}} \right|}}{{\left| {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  {{a_1}}&{{b_1}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}}
\end{array}} \right|}}$ Substitute $\left( {{x_1},{y_1},{z_1}} \right) \equiv \left( {5, - 2,0} \right)$; $\left( {{x_2},{y_2},{z_2}} \right) \equiv \left( {0,0,0} \right)$; $\left( {{a_1},{b_1},{c_1}} \right) \equiv \left( { - 2,0,1} \right)$; $\left( {{a_2},{b_2},{c_2}} \right) \equiv \left( {0,0,1} \right)$ in the shortest distance formula we get:
$
  D = \dfrac{{\left| {\begin{array}{*{20}{c}}
  {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\
  {{a_1}}&{{b_1}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}}
\end{array}} \right|}}{{\left| {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  {{a_1}}&{{b_1}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}}
\end{array}} \right|}} \\
   = \dfrac{{\left| {\begin{array}{*{20}{c}}
  {0 - 5}&{0 + 2}&{0 - 0} \\
  { - 2}&0&1 \\
  0&0&1
\end{array}} \right|}}{{\left| {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  { - 2}&0&1 \\
  0&0&1
\end{array}} \right|}} \\
   = \dfrac{{\left| { - 5(0 - 0) - 2( - 2 + 0) + 0} \right|}}{{\left| {\hat i(0) - \hat j( - 2) + \hat k(0 - 0)} \right|}} \\
   = \dfrac{{|0 + 4 + 0|}}{{|2\hat j|}} \\
   = \dfrac{4}{2} \\
   = 2{\text{ units}} \\
 $
Hence, the shortest distance between the z-axis and the line \[x + y + 2z - 3 = 0,2x + 3y + 4z - 4 = 0\] is 2 units.

Hence Option B is the correct answer.

Note:The distance from a point to a line is the shortest distance from a given point to any point on an infinite straight line. It is the perpendicular distance of the point to the line, the length of the line segment, which joins the point to the nearest point on the line. The formula for calculating it can be derived and expressed in several ways.
Knowing the distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on a graph, etc. In Deming regression, a type of linear curve fitting, if the dependent and independent variables have equal variance, this results in orthogonal regression in which the degree of imperfection of the fit is measured for each data point as the perpendicular distance of the point from the regression line.