Geometry • Topic 27

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. In geometry, they appear as loci of points with specific distance properties.

Definitions

Ellipse: Locus of points where sum of distances to two foci is constant ( $P F_{1} + P F_{2} = 2 a$ ).

2. Hyperbola: Locus where difference of distances is constant (

∣ P F_{1} - P F_{2} ∣ = 2 a

Parabola: Locus where distance to focus equals distance to directrix ( $PF = P D$ ).

[Image of Conic sections ellipse parabola hyperbola]

Reflection Property (Ellipse): A ray from one focus reflects off the ellipse to pass through the other focus.
Reflection Property (Parabola): Rays parallel to the axis reflect to pass through the focus.

All non-degenerate conics are equivalent under projective transformations.

Exercise (Problem 1)

Find the equation of the ellipse with foci at

(\pm 3, 0)

and major axis length 10.

Exercise (Problem 2)

Prove that the set of centers of circles tangent to a fixed circle and passing through a fixed point inside it is an ellipse.

Exercise (Problem 3)

Prove the reflection property of the parabola using calculus or geometry.