Conic Sections

Conic Sections

Jab ek double cone ko alag-alag angle par plane se kaata jaata hai, to jo curves milte hain — circle, parabola, ellipse, hyperbola — unhe conic sections kehte hain. Ye curves planets ki orbits se le kar satellite dishes tak har jagah dikhte hain.

1. Circle

Circle un saare points ka set hai jo ek fixed point (centre) se barabar doori (radius) par hon.

Centre \((h, k)\) aur radius \(r\) wale circle ka equation:

\[ (x - h)^{2} + (y - k)^{2} = r^{2}. \]

Centre origin par ho to: \(x^{2} + y^{2} = r^{2}\).

2. Parabola

Parabola un points ka set hai jo ek fixed point (focus) se aur ek fixed line (directrix) se barabar doori par hon.

Standard parabola (vertex origin par, right-opening):

\[ y^{2} = 4ax. \]

• Focus: \((a, 0)\), Directrix: \(x = -a\).
• Latus rectum ki length \(= 4a\).
• Other forms: \(y^{2} = -4ax\) (left), \(x^{2} = 4ay\) (up), \(x^{2} = -4ay\) (down).

3. Ellipse

Ellipse un points ka set hai jinke do fixed points (foci) se distances ka sum constant ho.

Standard ellipse (centre origin, \(a > b\)):

\[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1. \]

• Major axis length \(= 2a\), Minor axis length \(= 2b\).
• Foci: \((\pm c, 0)\), jahan \(c^{2} = a^{2} - b^{2}\).
• Eccentricity \(e = \dfrac{c}{a}\), aur \(0 < e < 1\).

4. Hyperbola

Hyperbola un points ka set hai jinke do foci se distances ka antar (difference) constant ho.

Standard hyperbola (centre origin):

\[ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1. \]

• Foci: \((\pm c, 0)\), jahan \(c^{2} = a^{2} + b^{2}\).
• Eccentricity \(e = \dfrac{c}{a}\), aur \(e > 1\).
• Transverse axis length \(= 2a\), Conjugate axis \(= 2b\).

5. Quick Comparison

Key Takeaways

• Circle: \((x-h)^{2}+(y-k)^{2}=r^{2}\).
• Parabola \(y^{2}=4ax\): focus \((a,0)\), latus rectum \(4a\).
• Ellipse: distances ka sum constant; \(c^{2}=a^{2}-b^{2}\), \(0
• Hyperbola: distances ka difference constant; \(c^{2}=a^{2}+b^{2}\), \(e>1\).
• Eccentricity \(e\) conic ka type batati hai.