Solveeit Logo
Login

Question

Question: If the image of the point \[P\left( {1, - 2,3} \right)\] in the plane \[2x + 3y - 4z + 22 = 0\] meas...

If the image of the point P(1,−2,3)P\left( {1, - 2,3} \right) in the plane 2x+3y−4z+22=02x + 3y - 4z + 22 = 0 measured parallel to the line x1=y4=x5\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{x}{5} is Q; the PQ is equal to:
A. 353\sqrt 5
B. 2422\sqrt {42}
C. 42\sqrt {42}
D.656\sqrt 5

Explanation

Solution

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=02x + 3y - 4z + 22 = 0, and the equation of line
x1=y4=z5\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{z}{5}
Equation of plane, P is
2x+3y−4z+22=02x + 3y - 4z + 22 = 0
Direction ratio of plane are (2,3,−4)\left( {2,3, - 4} \right)
Let the line be represented by L1{L_1}
Therefore, equation of line, L1{L_1}
x1=y4=z5\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{z}{5}
Therefore,
Direction ratio of line are (1,4,5)\left( {1,4,5} \right) equation of a line PQ passing through point P(1,−2,3)P\left( {1, - 2,3} \right) along line L1{L_1} is:

x−11=y+24=x−35=k ⇒x=k+1,y=4k−2,x=5k+3  \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(k+1)+3(4k+2)−4(5k+3)+22=0 ⇒k=1  2\left( {k + 1} \right) + 3\left( {4k + 2} \right) - 4\left( {5k + 3} \right) + 22 = 0 \\\ \Rightarrow k = 1 \\\

Coordinates of R=(2,2,8)R = \left( {2,2,8} \right)
Length of the image PQ = 2 PR.
Therefore,
PQ=2(2−1)2+(2+2)2+(8−3)2=242PQ = 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 xa=yb=zc,  a,b,c\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.