🔹 1. Introduction

Conic Sections are curves formed by the intersection of a plane with a double cone.

👉 Types:

• Circle
• Parabola
• Ellipse
• Hyperbola

These are important for:

• Coordinate Geometry
• JEE & CBSE exams
• Real-world applications (optics, astronomy)

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🔹 2. General Definition (Focus–Directrix)

A conic is the locus of a point such that:

👉 Ratio of distance from focus to distance from directrix is constant

This constant is called eccentricity (e).

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✔️ Classification Based on Eccentricity

• e = 0 → Circle
• e = 1 → Parabola
• e < 1 → Ellipse
• e > 1 → Hyperbola

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🔹 3. Circle

✔️ Definition

Set of all points at a constant distance from a fixed point (center).

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✔️ Standard Equation

$(x – h)^2 + (y – k)^2 = r^2$(x−h)2+(y−k)2=r2

$h$h

$k$k

$r$r

$(x)^2 + (y)^2 = 3.0^2$(x)2+(y)2=3.02-10-8-6-4-2246810-6-4-2246

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✔️ Key Terms

• Center = (h, k)
• Radius = r

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✔️ Special Case

$x^2 + y^2 = r^2$x2+y2=r2

$h$h

$k$k

$r$r

$(x)^2 + (y)^2 = 3.0^2$(x)2+(y)2=3.02-10-8-6-4-2246810-6-4-2246

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🔹 4. Parabola

✔️ Definition

Set of points equidistant from:

• Focus
• Directrix

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✔️ Standard Equation

$y^2 = 4ax$y2=4ax

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✔️ Key Elements

• Vertex = (0, 0)
• Focus = (a, 0)
• Directrix = x = −a

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✔️ Other Forms

• x² = 4ay
• y² = −4ax
• x² = −4ay

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🔹 5. Ellipse

✔️ Definition

Set of points where sum of distances from two foci is constant

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✔️ Standard Equation

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$a2x2​+b2y2​=1

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✔️ Important Relations

$c^2 = a^2 – b^2$c2=a2−b2

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✔️ Key Terms

• Center = (0, 0)
• Foci = (±c, 0)

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✔️ Eccentricity

$e = \frac{c}{a}$e=ac​

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🔹 6. Hyperbola

✔️ Definition

Set of points where difference of distances from two foci is constant

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✔️ Standard Equation

$\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1$a2x2​−b2y2​=1

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✔️ Important Relation

$c^2 = a^2 + b^2$c2=a2+b2

$a$a

$b$b

$c = \sqrt{a^2 + b^2} \approx 21.21$c=a2+b2​≈21.21

$a^2 + b^2 = c^2 \approx 225.00 + 225.00 = 450.00$a2+b2=c2≈225.00+225.00=450.00abc

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✔️ Eccentricity

$e = \frac{c}{a}$e=ac​

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✔️ Asymptotes

$y = \pm \frac{b}{a}x$y=±ab​x

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🔹 7. Rectangular Hyperbola

✔️ Equation

$xy = c^2$xy=c2

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🔹 8. Comparison Table

Conic	Equation	Eccentricity
Circle	x² + y² = r²	0
Parabola	y² = 4ax	1
Ellipse	x²/a² + y²/b² = 1	< 1
Hyperbola	x²/a² − y²/b² = 1	> 1

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🔹 9. Key Results

✔️ Circle is a special ellipse (a = b)
✔️ Parabola has one focus
✔️ Ellipse → sum constant
✔️ Hyperbola → difference constant

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🔹 10. Applications

✔️ Satellite dishes → Parabola
✔️ Planet orbits → Ellipse
✔️ Navigation systems → Hyperbola
✔️ Optics → Reflection properties

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🔹 11. Important Exam Points

✔️ Learn all standard equations
✔️ Focus-directrix concept is important
✔️ Practice identification of conics
✔️ Eccentricity-based questions are common
✔️ Hyperbola asymptotes frequently asked

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🔹 12. Common Mistakes

❌ Confusing ellipse & hyperbola formulas
❌ Sign errors (+ / −)
❌ Wrong values of a, b, c
❌ Ignoring orientation

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🔹 13. Practice Questions

1. Find equation of circle with center (2,3)
2. Find focus of parabola y² = 8x
3. Find eccentricity of ellipse
4. Identify conic from equation
5. Find asymptotes of hyperbola

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🔹 14. Quick Revision Sheet

• Circle → (x−h)² + (y−k)² = r²
• Parabola → y² = 4ax
• Ellipse → x²/a² + y²/b² = 1
• Hyperbola → x²/a² − y²/b² = 1
• Ellipse: c² = a² − b²
• Hyperbola: c² = a² + b²

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🔹 15. Conclusion

Conic Sections is a high-weightage and concept-based chapter in Class 11 Maths.

👉 Master:

• Standard forms
• Key formulas
• Properties

to score well in CBSE & JEE exams.