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نتیجہ
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How کیا مجھے classify a conic from Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0?
Use the discriminant Δ=B^2−4AC. If Δ<0: ellipse (A=C ⇒ circle), Δ=0: parabola, Δ>0: hyperbola.
How is the rotation angle chosen?
We use θ = ½·atan2(B, A−C) to eliminate the xy term, then round tiny residuals to zero for numerical stability.
How کیا مجھے obtain the standard form and پیرامیٹرز (a, b, p)?
After rotation, solve [2A' B'; B' 2C']·[X0;Y0] = −[D';E'] to find the center (or vertex). Translate to remove linear terms and normalize to circle/ellipse/hyperbola forms; for a parabola, complete the square to identify p in u^2=4pv.
How are foci, directrices, and asymptotes computed?
Ellipse: c=√(|a^2−b^2|), e=c/max(a,b); directrices are at ±a/e along the major axis. Hyperbola: c=√(a^2+b^2), e=c/a; asymptotes v=±(b/a)u mapped back after rotation/translation. Parabola u^2=4pv has focus (0,p) and directrix v=−p.
کیا میں شیئر کریں or export my نتائج?
Yes. Use URL کاپی کریں to شیئر کریں ان پٹ اقدار, LaTeX کاپی کریں for formulas, and CSV ایکسپورٹ کریں for sampled points from the plotted curve.