Doubtify – JEE Ki Taiyaari, Simple Bhasha Mein

❓ Question If the shortest distance between the lines x − 1 2 = y − 2 3 = z − 3 4 and x 1 = y α = z − 5 1 \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \quad\text{and}\quad \frac{x}{1}=\frac{y}{\alpha}=\frac{z-5}{1} is 5 6 , then the sum of all possible values of α \alpha α is equal to ? 🖼️ Question Image ✍️ Short Solution We’ll use the vector formula for the shortest distance between two skew lines : SD = ∣ ( b ⃗ − a ⃗ ) ⋅ ( d 1 ⃗ × d 2 ⃗ ) ∣ ∣ d 1 ⃗ × d 2 ⃗ ∣ where d ⃗ 1 , d ⃗ 2 \vec d_1,\vec d_2 are direction vectors and a ⃗ , b ⃗ \vec a,\vec b  are position vectors of any points on the lines. 🔹 Step 1 — Identify data from line equations Line 1 x − 1 2 = y − 2 3 = z − 3 4 \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} Point A ( 1 , 2 , 3 ) , d ⃗ 1 = ( 2 , 3 , 4 ) A(1,2,3),\quad \vec d_1=(2,3,4) Line 2 x 1 = y α = z − 5 1 \frac{x}{1}=\frac{y}{\alpha}=\frac{z-5}{1} Point (when parameter = 0): B ( 0 , 0 , 5 ) , d ⃗ 2 = ( 1 , α , 1 ) B(0,0,5),\quad \vec d_2=(1,\alpha...