If a line with direction ratio 2: 1: 1 intersects the lines (x-1/2)=(y-2/3)=(z-3/4) and ( x -2/1)=( y +1/2)=( z +2/3) at A and B then | AB | is

Question

If a line with direction ratio 2: 1: 1 intersects the lines (x-1/2)=(y-2/3)=(z-3/4) and ( x -2/1)=( y +1/2)=( z +2/3) at A and B then | AB | is (A) 2 √3 (B) 2 √5 (C) 2 √6 (D) 2 √7

Tardigrade Admin · Accepted Answer

A =(2 α+1,3 α+2,4 α+3) B =(β+2,2 β-1,3 β-2) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/a61cd422bfd835feb8d6deebd67e09ee-.png" /> Solving (I) (II) ⇒ 4 α-3 β+7=0 Solving (II) (III) ⇒ α-β+2=0 α=-1, β=1, A (-1,-1,-1) ; B (3,1,1) AB =2 √6

Q. If a line with direction ratio $2 : 1 : 1$ intersects the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{1} = \frac{y + 1}{2} = \frac{z + 2}{3}$ at $A$ and $B$ then $∣ A B ∣$ is

$23$

$25$

$26$

$27$

$A = (2 α + 1, 3 α + 2, 4 α + 3)$
$B = (β + 2, 2 β - 1, 3 β - 2)$

Solving (I) & (II) $\Rightarrow 4 α - 3 β + 7 = 0$
Solving (II) & (III) $\Rightarrow α - β + 2 = 0$
$α = - 1, β = 1, A (- 1, - 1, - 1); B (3, 1, 1)$
$A B = 26$

Q. If a line with direction ratio 2:1:1 intersects the lines 2x−1​=3y−2​=4z−3​ and 1x−2​=2y+1​=3z+2​ at A and B then ∣AB∣ is

23​

25​

26​

27​