find the lengthand equation to the line of s.d between lines x-3/1=y-5/-2=z-7/1
Answers
Step-by-step explanation:
MATHS
The equation of the line of shortest distance between the lines
4
x+4
=
−2
y−2
=
0
z−3
and
5
x−5
=
3
y−3
=
0
z
, is
Share
Study later
VIDEO EXPLANATION
ANSWER
Since, line of shortest distance is perpendicular to both the lines, its direction ratios can be obtained by cross-product of direction ratios of the two lines.
(4
i
^
−2
j
^
)×(5
i
^
+3
j
^
)=22
k
^
Direction of line of shortest distance=
k
^
Let
4
x+4
=
−2
y−2
=
0
z−3
=a
and
5
x−5
=
3
y−3
=
0
z
=b
Point of contact of first line and line of shortest distance =(4a−4,−2a+2,3)
Point of contact of second line and line of shortest distance =(5b+5,3b+3,0)
Since, line of shortest distance is perpendicular to both the lines,
4(4a−5b−9)−2(−2a−3b−1)=0⇒10a−7b=17
5(4a−5b−9)+3(−2a−3b−1)=0⇒7a−17b=24
On solving, we get a=1,b=−1
Substituting a=1, we get a point (0,0,3) that lies on the line of shortest distance
So, equation of line of shortest distance :
0
x
=
0
y
=
1
z−3