Question

A symmetrical form of the line of intersection of the planes \[x = ay + b,z = cy + d\] is
A. \[\dfrac{{x - b}}{a} = \dfrac{{y - 1}}{1} = \dfrac{{z - d}}{c}\]
B. \[\dfrac{{x - b - a}}{a} = \dfrac{{y - 1}}{1} = \dfrac{{z - d - c}}{c}\]
C. \[\dfrac{{x - a}}{b} = \dfrac{{y - 0}}{1} = \dfrac{{z - c}}{d}\]
D. \[\dfrac{{x - b - a}}{a} = \dfrac{{y - 1}}{0} = \dfrac{{z - d - c}}{d}\]

Answer 1

Hint:Here we write both the equations of the plane and by shifting the terms we write both equations in a way that they equate to y on one side, and then we write the values equal to y as equal to each other. This will give us a line of intersection of two planes which will be in a fraction form but all fractions are symmetric because they are equal to each other.

Complete step-by-step answer:
We have equation of plane as \[x = ay + b\]
Shift all values other than with variable y to one side of the equation
\[ \Rightarrow x - b = ay\]
Divide both sides by a
\[ \Rightarrow \dfrac{{x - b}}{a} = \dfrac{{ay}}{a}\]
Cancelling same terms from numerator and denominator
\[ \Rightarrow \dfrac{{x - b}}{a} = y\] … (1)
Now have equation of plane as \[z = cy + d\]
Shift all values other than with variable y to one side of the equation
\[ \Rightarrow z - d = cy\]
Divide both sides by c
\[ \Rightarrow \dfrac{{z - d}}{c} = \dfrac{{cy}}{c}\]
Cancelling same terms from numerator and denominator
\[ \Rightarrow \dfrac{{z - d}}{c} = y\] … (2)
Now, since RHS of both equations (1) and (2) is same, we can write
\[\dfrac{{x - b}}{a} = y = \dfrac{{z - d}}{c}\]
Now, write the term with y as \[\dfrac{{y - 0}}{1}\]
\[\dfrac{{x - b}}{a} = \dfrac{{y - 0}}{1} = \dfrac{{z - d}}{c}\]
There is no option in the question that matches the equation, so we will transform the equation by subtracting 1 from all values
\[ \Rightarrow \dfrac{{x - b}}{a} - 1 = \dfrac{{y - 0}}{1} - 1 = \dfrac{{z - d}}{c} - 1\]
By taking LCM in each term
\[ \Rightarrow \dfrac{{x - b - a}}{a} = \dfrac{{y - 1}}{1} = \dfrac{{z - d - c}}{c}\]

So, the correct answer is “Option B”.

Note:Students might make mistake of using substitution method to find the symmetrical form of intersecting line but that will be very complex to solve as all the values will be combined first and then we have to break them, also many students might select option A but that is wrong because in numerator we don’t have the value y-1.