• 3D Line Through a Point & Two Lines — JEE Trick in 59 Seconds! 🔥

 

❓ 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.

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✍️ 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!

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🔹 Step 1 — Write Given Lines in Parametric Form

Suppose:

$$\vec r = \vec a_1 + \lambda \vec d_1$$
$$\vec r = \vec a_2 + \mu \vec d_2$$

Where

• $\vec a_1,\ \vec a_2$ = position vectors of known points

• $\vec d_1,\ \vec d_2$ = direction vectors

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🔹 Step 2 — Find Intersection Points with L

Assume the line L passes through a fixed point $P$.

Let

$$A = \vec a_1 + \lambda \vec d_1$$
$$B = \vec a_2 + \mu \vec d_2$$

Since L intersects both lines,
👉 A and B both lie on L

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🔹 Step 3 — Use Collinearity with P (Key Step 🔥)**

Points P, A, B are collinear, so:

$$\vec{A} - \vec{P}$$

is parallel to

$$\vec{B} - \vec{P}$$

So we write:

$$\vec{A} - \vec{P} = k(\vec{B} - \vec{P})$$

Compare x, y, z components
→ you get equations in $\lambda,\ \mu,\ k$

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🔹 Step 4 — Solve for Parameters

From the three component equations:

✔ Solve for $\lambda$
✔ Solve for $\mu$
✔ (Optional) get $k$

Now you have coordinates of A or B explicitly.

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🔹 Step 5 — Final Line & Checking a Point

Direction vector of line L:

$$\vec{d_L} = \vec{A} - \vec{P} \quad \text{(or } \vec{B} - \vec{P})$$

Equation of L:

$$\vec r = \vec P + t\vec{d_L}$$

To check if any option lies on L:

✔ Put its $(x,y,z)$
✔ See if the same t satisfies all three equations

👉 If yes → point lies on L

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✅ Final Takeaway

🧠 3D Master Trick

1️⃣ Convert lines to parametric form
2️⃣ Assume intersection points $A,B$
3️⃣ Apply collinearity with $P$
4️⃣ Solve parameters
5️⃣ Form final line & verify options

Is flow ko yaad rakho —
👉 3D “line through a point & cutting two lines” questions ho jaate hain 1-min solve!