To determine if the given straight lines intersect, we need to find a common point of intersection that satisfies both parametric equations. The lines are given by:
===k1where
k1 is a parameter, and:
===k2 where
k2 is another parameter.
First, express the coordinates
(x,y,z) in terms of
k1 and
k2 :
From the first line:
x=2+k1y=3+k1z=4−tk1 From the second line:
x=1+tk2y=4+2k2z=5+k2 For the lines to intersect, there must exist values of
k1 and
k2 such that the corresponding coordinates are equal. Thus, we set up the following system of equations:
For
x :
2+k1=1+tk2For
y :
3+k1=4+2k2For
z :
4−tk1=5+k2 We now solve this system of equations:
From the second equation, we have:
k1−2k2=1k1=1+2k2 Substitute
k1=1+2k2 into the first equation:
2+1+2k2=1+tk23+2k2=1+tk22+2k2=tk2tk2−2k2=2k2(t−2)=2k2= Substitute
k1=1+2k2 and
k2= into the third equation:
To simplify, multiply both sides by
t−2 :
4(t−2)−t(t−2)−4t=5(t−2)+2
4t−8−t2+2t−4t=5t−10+2
4t−t2−2t−8=5t−8−t2−2t=5t−t2−7t=0−t(t+7)=0t=0 or t=−7 So, the lines intersect for exactly two values of
t.
Hence, the correct option is:
Option B: Exactly two values