Question

Show that the lines x+3/-3 ​= y−1/1 ​= z−5/5​ and x+1/-1 ​= y−2/2 ​= z−5/5​ are coplanar. Also find the equation of the plane.

Answer 1

If two lines are coplanar then

Given (x₁​,y₁​,z₁​) is (−3,1,5) and (x₂​,y₂​,z₂​) is (−1,2,5) and l₁​,m₁​,n₁​=−3,1,5

l₂​,m₂​,n₂​=−1,2,5

Substituting the values

2(5−10)−1(−15+5)=0

Hence the lines are coplanar.

The equation of the plane containing these lines is

(x+3)(5−10)−(y−1)(−15+5)+(2−5)(−6+1)

⇒−5(x+3)+10(y−1)−5(z−5)

⇒−5x+10y−5z−30=0

5x−10y+5z+30=0

5(x−2y+z+6)=0

x−2y+z+6=0

or

5x−10y+5z+30=0

∴ The equation of the plane is 5x−10y+5z+30=0.