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# Solution - Find the shortest distance between the lines (x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-4)/4=(z-5)/5 - HSC Science (General) 12th Board Exam - Mathematics and Statistics

ConceptPair of Lines Passing Through Origin - Combined Equation

#### Question

Find the shortest distance between the lines (x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-4)/4=(z-5)/5

#### Solution

The lines are

(X-1)/2=(Y-2)/3=(Z-3)/4................(1)

(x-2)/3=(y-4)/4=(z-5)/5.................(2)

Here

x_1=1,y_1=2,z_1=3 and  a_1=2,b_1=3,c_1=4

x_2=2,y_2=4,z_2=5 and  a_2=3,b_2=4,c_2=5

Shortest distance between the lines is

d=|[x_2-x_1,y_2-y_1,z_2-z_1],[a_1,b_1,c_1],[a_2,b_2,c_2]|/sqrt((b_1c_2-b_2c_1)^2+(c_1a_2-c_2a_1)^2+(a_1b_2-a_2b_1)^2)

Now |[x_2-x_1,y_2-y_1,z_2-z_1],[a_1,b_1,c_1],[a_2,b_2,c_2]|=|[1,2,2],[2,3,4],[3,4,5]|

=1(15-16)-2(10-12)+2(8-9)=-1+4-2=1

and (b_1c_2-b_2c_1)^2+(c_1a_2-c_2a_1)^2+(a_1b_2-a_2b_1)^2=(15-16)^2+(12-10)^2+(8-9)^2

=1+4+1=6

Hence, the shortest distance between the lines (i) and (ii) is=|1/sqrt6|=1/sqrt6 units

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#### APPEARS IN

2016-2017 (March) (with solutions)
Question 2.1.2 | 3 marks
Solution for question: Find the shortest distance between the lines (x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-4)/4=(z-5)/5 concept: null - Pair of Lines Passing Through Origin - Combined Equation. For the courses HSC Science (General) , HSC Science (Computer Science), HSC Arts, HSC Science (Electronics)
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