Question

If the image of the point \[P\left( {1, - 2,3} \right)\] in the plane \[2x + 3y - 4z + 22 = 0\] measured parallel to the line \[\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{x}{5}\] is Q; the PQ is equal to:
A. \[3\sqrt 5 \]
B. \[2\sqrt {42} \]
C. \[\sqrt {42} \]
D.\[6\sqrt 5 \]

Answer 1

Hint: The coordinate of the point is the place from where the line passes to meet the stated condition. A plane is a two-dimensional analogue of a point, a line and three-dimensional space. If a, b, c are three numbers proportional to the direction cosine l, m, n of a straight line, then a, b, c is called direction rations.

Complete step by step solution:
Given: The equation of plane is
\[2x + 3y - 4z + 22 = 0\], and the equation of line
\[\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{z}{5}\]
Equation of plane, P is
\[2x + 3y - 4z + 22 = 0\]
Direction ratio of plane are \[\left( {2,3, - 4} \right)\]
Let the line be represented by \[{L_1}\]
Therefore, equation of line, \[{L_1}\]
\[\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{z}{5}\]
Therefore,
Direction ratio of line are \[\left( {1,4,5} \right)\] equation of a line PQ passing through point \[P\left( {1, - 2,3} \right)\] along line \[{L_1}\] is:
\[
  \dfrac{{x - 1}}{1} = \dfrac{{y + 2}}{4} = \dfrac{{x - 3}}{5} = k \\
   \Rightarrow x = k + 1, y = 4k - 2, x = 5k + 3 \\
 \]
Let's say R is the midpoint of the line PQ, therefore R lies on the plane.
Hence it should satisfy the equation of the plane
\[
  2\left( {k + 1} \right) + 3\left( {4k + 2} \right) - 4\left( {5k + 3} \right) + 22 = 0 \\
   \Rightarrow k = 1 \\
 \]
Coordinates of \[R = \left( {2,2,8} \right)\]
Length of the image PQ = 2 PR.
Therefore,
\[PQ = 2\sqrt {{{\left( {2 - 1} \right)}^2} + {{\left( {2 + 2} \right)}^2} + {{\left( {8 - 3} \right)}^2}} = 2\sqrt {42} \]
Thus option (2) is correct.

Note: In this type of question students often make mistakes in determining the plane and choosing the direction ratios. Remember the direction cosines are different from direction ratio and that they are not the same. In the equation of line \[\dfrac{x}{a} = \dfrac{y}{b} = \dfrac{z}{c},\,\,a,b,c\] are the direction ratios and not direction cosines keep these points in mind before solving the question.