General second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0 — discriminant classification

3.4.11 · Maths › Conic Sections

YE equation HAI KYA?

$B^2-4AC$ shape KYU decide karta hai?

Ye magic isliye hai kyunki $B^2-4AC$ ek rotation invariant hai: axes ko rotate karo taaki $xy$ term khatam ho jaye, aur ye number nahi badalta. To ye shape ki koi intrinsic cheez capture karta hai, tilt ki nahi.

First principles se derivation — rotation invariance

Step 1 — Axes ko angle $\theta$ se rotate karo. Replace karo $x=x^{\prime }\cos \theta -y^{\prime }\sin \theta ,y=x^{\prime }\sin \theta +y^{\prime }\cos \theta .$ Ye step kyun? Ek tilted conic kisi rotated frame mein axis-aligned ho jaati hai; hum woh frame dhundhna chahte hain.

Step 2 — $A{x}^2+Bxy+C{y}^2$ mein substitute karo. Terms collect karne par naye coefficients $A^{\prime },B^{\prime },C^{\prime }$ milte hain: $A^{\prime }=A{\cos }^2\theta +B\cos \theta \sin \theta +C{\sin }^2\theta ,$ $C^{\prime }=A{\sin }^2\theta -B\cos \theta \sin \theta +C{\cos }^2\theta ,$ $B^{\prime }=B\cos 2\theta -(A-C)\sin 2\theta .$ Ye step kyun? Hum dekhna chahte hain ki $A,B,C$ kaise transform hote hain, taaki ek aisi quantity dhundh sakein jo fixed rahe.

Step 3 — $\theta$ choose karo taaki cross term eliminate ho. $B^{\prime }=0$ set karo: $B\cos 2\theta =(A-C)\sin 2\theta \Rightarrow \tan 2\theta =\frac{B}{A-C}.$ Ye step kyun? Jab $B^{\prime }=0$ ho jata hai, equation mein $xy$ nahi hota: ye ek standard, un-tilted conic hai jise hum instantly pehchan lete hain.

Step 4 — $B^{\prime 2}-4A^{\prime }{C}^{\prime }$ compute karo. Algebra ke baad (neeche verified): $B^{\prime 2}-4A^{\prime }{C}^{\prime }=B^2-4AC.$ Ye step kyun? Ye prove karta hai ki $B^2-4AC$ rotation ke under invariant hai — ye shape ki property hai, tilt ki nahi.

Step 5 — Rotated frame mein ($B^{\prime }=0$) shape padhho. Equation aisa dikhta hai $A^{\prime }{x}^{\prime 2}+C^{\prime }{y}^{\prime 2}+\cdots =0.$

• $A^{\prime }{C}^{\prime }>0$ (same sign): $A^{\prime }{x}^{\prime 2}+C^{\prime }{y}^{\prime 2}=\text{const}$ → ellipse. Tab $B^{\prime 2}-4A^{\prime }{C}^{\prime }=-4A^{\prime }{C}^{\prime }<0$. ✓
• $A^{\prime }{C}^{\prime }<0$ (opposite signs): → hyperbola. Tab $-4A^{\prime }{C}^{\prime }>0$. ✓
• $A^{\prime }{C}^{\prime }=0$ (ek zero hai): sirf ek squared term → parabola. Tab $-4A^{\prime }{C}^{\prime }=0$. ✓

Kyunki $B^2-4AC=B^{\prime 2}-4A^{\prime }{C}^{\prime }=-4A^{\prime }{C}^{\prime }$, $B^2-4AC$ ka sign hume $A^{\prime }{C}^{\prime }$ ka sign batata hai — isliye shape pata chalta hai. QED.

POORI classification KAISE karein (degenerate cases samet)

Ek conic collapse ho sakti hai: ellipse ek point mein, hyperbola do crossing lines mein, parabola parallel lines mein. Inhe pakadne ke liye full $3\times 3$ determinant use karo: $\Delta =\left|\begin{array}{ccc}A& B/2& D/2\\ B/2& C& E/2\\ D/2& E/2& F\end{array}\right|.$

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

• Rotation of Axes — woh transformation jo $xy$ term remove karta hai.
• Quadratic Forms and Eigenvalues — $B^2-4AC$ ka sambandh $\left(\begin{array}{cc}A& B/2\\ B/2& C\end{array}\right)$ ke eigenvalues ke sign se hai.
• Ellipse Standard Equation, Parabola Standard Equation, Hyperbola Standard Equation — axis-aligned targets.
• Circle as Special Ellipse — $A=C,B=0$.
• Discriminant of a Quadratic — 1-variable cousin $b^2-4ac$ (confuse mat karo!).
• Degenerate Conics — point, line, line-pairs jab $\Delta =0$.

Concept Map