Hyperbola Focus & Latus Rectum Trick in 59 Seconds! 🔥 | JEE Maths

❓ Concept

🎬 Hyperbola – Focus & Latus Rectum Concept in 59 Sec

Hyperbola dikhe, focus dikhe,
aur latus rectum right angle subtend kar raha ho?

👉 Samajh jao — geometry ka gold mine mil chuka hai 🔥
Yahan formula kam, logic zyada kaam karta hai.

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🖼️ Concept Image

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✍️ Short Explanation

Is type ke questions JEE mein isliye favourite hain kyunki:
✔ Algebra aur geometry perfectly combine hote hain
✔ Ek right-angle condition poora problem lock kar deta hai

Agar steps clear hain, toh answer clean aur fast nikalta hai.

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🔹 Step 1 — Standard Hyperbola Basics (FOUNDATION 💯)**

Standard form:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

Is hyperbola ke liye:

$$c^2=a^2+b^2$$
$$\text{Foci}=(\pm c,0)$$

📌 Agar focus diya ho, toh c directly mil jaata hai —
aur yahin se poora question start hota hai.

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🔹 Step 2 — Latus Rectum of Hyperbola

Hyperbola ka latus rectum:

• Har focus se guzarta hai

• Transverse axis ke perpendicular hota hai

Equation:

$$x=\pm c$$

Endpoints:

$$\left(\pm c,\ \pm\frac{b^2}{a}\right)$$

Length:

$$\text{Latus rectum}=\frac{2b^2}{a}$$

📌 Yeh coordinates exam mein direct use hote hain.

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🔹 Step 3 — Right Angle Subtended at a Point (GAME CHANGER 🔥)**

Agar latus rectum ke endpoints
$L_1$ aur $L_2$
kisi point $P$ par 90° subtend karein:

Two equivalent methods:

• Slope method:

$$(\text{slope of }P{L}_1)\times (\text{slope of }P{L}_2)=-1$$

• Vector method (cleaner 😎):

$$\overrightarrow{P{L}_1}\cdot \overrightarrow{P{L}_2}=0$$

👉 Yehi ek condition $a$ aur $b$ ke beech
solid relation de deti hai.

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🔹 Step 4 — Focus Given ⇒ Hyperbola Fixed

Question jab bole:

“One focus is at $(x_0,0)$”

Toh:

$$c=\mathrm{\mid }{x}_0\mathrm{\mid }\Rightarrow a^2+b^2=c^2$$

📌 Ab:

• Ek equation geometry se

• Ek equation right-angle condition se

👉 Solve karo → $a, b$ mil jaate hain.

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🔹 Step 5 — Why JEE Asks $a^2b^2$ (SMART OBSERVATION 🧠)**

JEE often poochta hai:

$$a^2{b}^2=\text{something}$$

Reason:

• $a^2b^2$ symmetric expression hai

• Square roots hat jaate hain

• Final answer clean form mein aata hai

Once $a$–$b$ relation mil gaya:

$$a^2{b}^2\Rightarrow \text{direct computation}\text{<br>}\text{<br>}$$

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✅ Final Takeaway

🧠 Hyperbola Gold Rules

• Focus ⇒ $c$ known

• Hyperbola identity ⇒ $a^2+b^2=c^2$
  

• Latus rectum endpoints ⇒ $(\pm c,\pm b^2/a)$
  

• Right angle ⇒ dot product = 0

• JEE loves ⇒ $a^2b^2$
  

Is concept ko samajh liya,
👉 Hyperbola ke tough-looking questions bhi 1 minute mein khatam 🚀