Doubtify – JEE Ki Taiyaari, Simple Bhasha Mein

  ❓ Concept 🎬 3D Line Through a Point & Two Lines JEE 3D geometry mein ek super-common pattern hota hai: 👉 A line L passes through a fixed point and intersects two given lines. Iska matlab hai — L ko define karne ke liye bas parameter aur collinearity ka game khelo. ✍️ Short Explanation Trick simple hai: ✔ Dono lines ki parametric form likho ✔ Assume intersection points ✔ Use collinearity with the fixed point ✔ Solve parameters ✔ Get direction of L → equation ready! 🔹 Step 1 — Write Given Lines in Parametric Form Suppose: r ⃗ = a ⃗ 1 + λ d ⃗ 1 \vec r = \vec a_1 + \lambda \vec d_1 r ⃗ = a ⃗ 2 + μ d ⃗ 2 \vec r = \vec a_2 + \mu \vec d_2 Where a ⃗ 1 ,   a ⃗ 2 \vec a_1,\ \vec a_2  = position vectors of known points d ⃗ 1 ,   d ⃗ 2 \vec d_1,\ \vec d_2 ​ = direction vectors 🔹 Step 2 — Find Intersection Points with L Assume the line L passes through a fixed point P P P . Let A = a ⃗ 1 + λ d ⃗ 1 A = \vec a_1 + \lambda \vec d_1 B = ...

❓ Question Let the line L pass through the point ( 1 ,   1 ,   1 ) (1,\,1,\,1) and intersect the lines x − 1 2 = y + 1 3 = z − 1 4 and x − 3 1 = y − 4 2 = z 1 . \frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4} \quad \text{and} \quad \frac{x-3}{1} = \frac{y-4}{2} = \frac{z}{1}. Which of the following points lies on the line L ? 🖼️ Question Image ✍️ Short Solution The idea is very standard in JEE 3D geometry : 👉 If a line L intersects two given lines , then the two intersection points together determine line L . 👉 Once the equation of L is known, we simply check which option satisfies it. 🔹 Step 1 — Write parametric form of the first line Given: x − 1 2 = y + 1 3 = z − 1 4 = λ \frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4} = \lambda So, x = 1 + 2 λ , y = − 1 + 3 λ , z = 1 + 4 λ x = 1 + 2\lambda,\quad y = -1 + 3\lambda,\quad z = 1 + 4\lambda Let the point of intersection with L be: P ( 1 + 2 λ ,   − 1 + 3 λ ,    1 + 4 λ ) P(1+2\lambda,\,-1+3\lambda,\;1+4\lambda) 🔹 ...