Conic Sections 🔵

Ek cone ko alag angles par kaato — circle, ellipse, parabola, hyperbola milte hain! Nature aur engineering mein everywhere hain yeh curves.

🌍 Real World mein Conics

Yeh ek amazing fact hai — ek double cone ko alag angle par katne se 4 alag shapes milte hain. Circle: wheels, plates, pizza! Ellipse: Earth sun ke around elliptical orbit mein ghoomti hai. Parabola: Cricket ball ka trajectory (air resistance ignore karo), satellite dish ka shape. Hyperbola: Ek lamp se wall par light ka shadow. In saari shapes ka ek general equation hai — aur yahi hum sikhenge!

Double Cone ko alag angles par kaatne se milte hain 4 conics:

Circle horizontal cut Ellipse tilted cut Parabola parallel to side Hyperbola steep cut (2 parts)

1. Circle

Circle: Ek fixed point (centre) se ek fixed distance (radius) par ke saare points ka set. Simple aur symmetric — sabse easy conic!

Standard form: x² + y² = r² (centre at origin)
General form: (x−h)² + (y−k)² = r² (centre at (h,k))

General equation x²+y²+2gx+2fy+c=0 mein: centre = (−g,−f), radius = √(g²+f²−c)

✅ Circle ka equation: centre (3,−2), radius 5

(x−3)² + (y+2)² = 25
Expand karo: x²−6x+9 + y²+4y+4 = 25
x² + y² − 6x + 4y − 12 = 0

✅ x²+y²−4x+6y−3=0 ka centre aur radius nikalo

Complete the square: (x²−4x+4) + (y²+6y+9) = 3+4+9
(x−2)² + (y+3)² = 16
Centre = (2,−3), Radius = 4

2. Parabola

Parabola: Ek fixed point (focus F) aur ek fixed line (directrix) se equal distance par ke saare points. Yeh symmetrical U-shape curve hai jisme ek axis of symmetry hoti hai.

a = focus ki origin se distance. Vertex = origin par (standard form mein).

y² = 4ax (rightward, focus at (a,0), directrix x=−a)
y² = −4ax (leftward)
x² = 4ay (upward, focus at (0,a))
x² = −4ay (downward)
F(1,0) x=−1 Vertex(0,0) x-axis Axis of symmetry y²=4x

3. Ellipse

Ellipse: Do fixed points (foci F₁,F₂) se milne wali distances ka sum constant rehta hai. Circle ka generalised version — ek axis dusre se bada hota hai. a = semi-major axis, b = semi-minor axis, c = focus ki centre se distance.

x²/a² + y²/b² = 1 (horizontal major axis, a > b)
x²/b² + y²/a² = 1 (vertical major axis)
Relation: c² = a² − b² | Eccentricity: e = c/a (0 < e < 1)

e=0 → circle, e closer to 1 → more elongated ellipse. Earth ka e ≈ 0.017 (nearly circular orbit!)

✅ x²/25 + y²/16 = 1 ke liye foci, vertices, eccentricity nikalo

a² = 25, b² = 16 → a = 5, b = 4 (a > b → horizontal major axis)
c² = a²−b² = 25−16 = 9 → c = 3
Foci: (±3, 0), Vertices: (±5, 0) aur (0, ±4)
Eccentricity e = c/a = 3/5 = 0.6

4. Hyperbola

Hyperbola: Do fixed points (foci) se milne wali distances ka difference constant rehta hai (sum nahi!). Do branches milti hain jo opposite directions mein jaati hain — asymptotes ke close aati jaati hain.

x²/a² − y²/b² = 1 (horizontal transverse axis)
y²/a² − x²/b² = 1 (vertical transverse axis)
c² = a² + b² (notice: + not −, unlike ellipse!)
Asymptotes: y = ±(b/a)x

5. Comparison Table — Saare Conics

ConicStandard EquationKey FeatureEccentricity e
Circlex²+y²=r²All points equidistant from centree = 0
Parabolay²=4ax1 focus, 1 directrix, equal distancee = 1
Ellipsex²/a²+y²/b²=1Sum of distances to foci = 2a0 < e < 1
Hyperbolax²/a²−y²/b²=1Difference of distances = 2ae > 1

6. Practice Questions

Q1. Circle x²+y²−6x+8y+9=0 ka centre aur radius nikalo.

Solution

(x−3)²+(y+4)² = 9+16−9 = 16. Centre=(3,−4), radius=4

Q2. Parabola y²=12x ke focus aur directrix nikalo.

Solution

4a=12 → a=3. Focus=(3,0), Directrix: x=−3

Q3. Ellipse x²/36+y²/20=1 ke foci aur eccentricity nikalo.

Solution

a²=36, b²=20, c²=16 → c=4. Foci=(±4,0), e=4/6=2/3

Q4. Hyperbola x²/9−y²/16=1 ke asymptotes likho.

Solution

y = ±(b/a)x = ±(4/3)x

Q5. Focus (2,0) aur directrix x=−2 wali parabola ka equation likho.

Solution

a=2 → y² = 4(2)x = y²=8x

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