JEE Main: Smart Method to Solve Line Through a Point & Two Lines 💡
❓ 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) 🔹 ...