Conic Sections

Board: CBSE | Class: Class 11

Comprehensive study notes for Conic Sections by Ajay Yadav (Math King of Katargam).

Key Concepts

Conic Sections

Sections of a double cone by a plane: Circle (plane ⊥ axis), Parabola (plane parallel to generator), Ellipse (plane angled to axis), Hyperbola (plane steeper than generator).

Circle

(x-h)² + (y-k)² = r² where (h,k) is center, r is radius. General form: x²+y²+2gx+2fy+c=0, center (-g,-f), r = √g²+f²-c.

Parabola

Set of points equidistant from focus and directrix. y² = 4ax (opens right). Focus: (a,0). Directrix: x=-a. Latus rectum: 4a. Other forms: y²=-4ax, x²=4ay, x²=-4ay.

Ellipse

Set of points where sum of distances to two foci is constant. x²/a² + y²/b² = 1 (a>b). Foci: (±c,0), c² = a²-b². Eccentricity e = c/a (<1).

Hyperbola

Set of points where difference of distances to two foci is constant. x²/a² – y²/b² = 1. Foci: (±c,0), c² = a²+b². Eccentricity e = c/a (>1). Asymptotes: y = ±(b/a)x.

Important Formulas

Circle	(x-h)²+(y-k)² = r²
Parabola (right)	y² = 4ax, focus (a,0)
Ellipse	x²/a² + y²/b² = 1, e = √1-b²/a²
Hyperbola	x²/a² - y²/b² = 1, e = √1+b²/a²

Solved Examples

Example 1: Find center and radius of x²+y²-6x+8y-11=0.

Solution: Complete squares: (x-3)²+(y+4)²=36. Center: (3,-4), radius: 6.

Example 2: Find focus and directrix of y²=12x.

Solution: y² = 4(3)x ⇒ a=3. Focus: (3,0). Directrix: x=-3.

Example 3: Find eccentricity of ellipse x²/25+y²/9=1.

Solution: a²=25, b²=9. c²=25-9=16, c=4. e = 4/5 = 0.8.

Practice Questions

1. Find equation of circle with center (2,-3) and radius 5.
2. Find focus, vertex, directrix of y²=-8x.
3. For ellipse x²/16+y²/9=1, find foci and eccentricity.
4. For hyperbola x²/9-y²/16=1, find asymptotes.
5. Find the latus rectum of parabola y²=16x.

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