Question

The equation of plane in which the lines $ \dfrac{{x - 5}}{4} = \dfrac{{y - 7}}{4} = \dfrac{{z + 3}}{{ - 5}} $ and $ \dfrac{{x - 8}}{7} = \dfrac{{y - 4}}{1} = \dfrac{{z - 5}}{3} $ lie is:
a) $ 17x - 47y - 24z + 172 = 0 $
b) $ 17x + 47y - 24z + 172 = 0 $
c) $ 17x + 47y + 24z + 172 = 0 $
d) $ 17x - 47y + 24z + 172 = 0 $

Answer 1

Hint: We are given two line equations and are asked to find a plane that contains these two lines. We will assume a general plane equation and use the d.r.s of one line and point of another line to generate two equations with three unknowns. On solving that we will get the answer.
Formula used:
1) The general equation of a plane is $ ax + by + cz + d = 0 $
2) If a line having drs l,m,n lies in the plane $ ax + by + cz + d = 0 $ then,
 $ al + bm + cn = 0 $

Complete step-by-step answer:
Let $ L:\dfrac{{x - 5}}{4} = \dfrac{{y - 7}}{4} = \dfrac{{z + 3}}{{ - 5}} $ and $ L':\dfrac{{x - 8}}{7} = \dfrac{{y - 4}}{1} = \dfrac{{z - 5}}{3} $
Let $ ax + by + cz + d = 0 $ be a plane that contains L and L’
A point on the line L is $ (5,7, - 3) $
So the equation of plane passing through the line L is
 $ a(x - 5) + b(y - 7) + c(z + 3) = 0 $
Now d.c.s of L is 4,4,-5
As d.c.s of the required plane is a,b,c and it contains L so we have,
 $ 4a + 4b - 5c = 0 $ --(1)
A point on L’ is $ (8,4,5) $
Our assumed plane will pass through the second line L’ if $ (8,4,5) $ lies on it.
That is if $ a(8 - 5) + b(4 - 7) + c(5 + 3) = 0 $
 $ \Rightarrow 3a - 3b + 8c = 0 $ --(2)
From (1) and (2) we have,
 $ 4a + 4b - 5c = 0 $
 $ 3a - 3b + 8c = 0 $
On solving we get,
 $ \Rightarrow \dfrac{a}{{32 - 15}} = \dfrac{{ - b}}{{32 + 15}} = \dfrac{c}{{ - 12 - 12}} $
\[ \Rightarrow \dfrac{a}{{17}} = \dfrac{{ - b}}{{47}} = \dfrac{c}{{ - 24}}\]
\[ \Rightarrow \dfrac{a}{{17}} = \dfrac{b}{{ - 47}} = \dfrac{c}{{ - 24}} = k(say)\]
So we can write $ a = 17,b = - 47,c = - 24 $
As our plane equation is $ ax + by + cz + d = 0 $
We can write it as:
 $ 17x - 47y - 24z + d = 0 $
As it passes through $ (8,4,5) $ so we have,
 $ 17 \times 8 - 47 \times 4 - 24 \times 5 + d = 0 $
\[ \Rightarrow d = 172\]
Now we have the plane equation as,
 $ 17x - 47y - 24z + 172 = 0 $
As this equation is identical to option ‘a’ so it is the correct option.
So, the correct answer is “Option A”.

Note: Always use the cross ratio technique to solve a system of linear equations with two equations containing three unknowns. It is the easiest and quickest method to solve it. The cross ratio technique involves cross multiplication of coefficients of the unknown variables.