JEE Main: Parabola from Focus–Directrix — Fast & Simple 💡

 

❓ Concept

🎬 Parabola: Focus–Directrix Trick

Parabola ka equation yaad nahi?
👉 Koi tension nahi!

Bas focus–directrix definition yaad rakho —
aur poora parabola khud-ba-khud ban jaata hai.

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🖼️ Concept Image

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✍️ Short Explanation

Parabola ek distance-based curve hai.
Equation derivation ya formula yaad karne ki zarurat hi nahi —
sirf distance equality lagao, square karo, equation mil jaata hai.

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🔹 Step 1 — Definition of Parabola

Parabola = locus of a point P(x, y) such that

$$\text{Distance(P, Focus)}=\text{Distance(P, Directrix)}$$

Bas — yeh hi official definition hai 🔥

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🔹 Step 2 — Distance Formula Concept

✔ Distance from focus
→ point-to-point distance using

$$\sqrt{(x-x_1{)}^2+(y-y_1{)}^{\mathrm{2}}}$$

✔ Distance from directrix
→ perpendicular distance from a line

$$\frac{\mathrm{\mid }ax+by+c\mathrm{\mid }}{\sqrt{a^2+b^{\mathrm{2}}}}$$

👉 In dono ko equal rakhte hain.

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🔹 Step 3 — General Method (Always Works 💯)**

Let $P(x,y)$ lie on parabola.

Write:

$$\text{Distance(P, Focus)} = \text{Distance(P, Directrix)}$$

Now:

1️⃣ Square both sides (square-root hat jaata hai)
2️⃣ Expand and simplify
3️⃣ Equation of parabola mil jaati hai

Isse standard form yaad karne ki zarurat nahi.

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🔹 Step 4 — Fixing One Coordinate (Smart Trick 😎)**

Agar question bole:

👉 “Points on parabola whose abscissa = constant”

Matlab $x = k$

Toh simply:

• $x = k$ put karo

• Equation quadratic in $y$ ban jayegi

• Matlab multiple y-values possible (2 points usually)

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🔹 Step 5 — Sum of Ordinates Trick (Fastest JEE Shortcut 🔥)**

Quadratic in $y$:

$$ay^2 + by + c = 0$$

Then:

$$\text{Sum of roots} = -\frac{b}{a}$$

👉 Yeh direct formula use karo — roots nikaalne ki zarurat nahi.

Isse sum of ordinates seconds mein mil jaata hai.

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✅ Final Takeaway

🧠 Parabola Master Rules

• Parabola = distance from focus = distance from directrix

• Distance formulas apply → square → equation ready

• $x=k$ fix karne par → quadratic in $y$
  

• Sum of ordinates = −b/a (quadratic shortcut)

Isse focus–directrix wale saare JEE questions ultra-easy ho jaate hain 🚀

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