Conic sectioins

CONIC SECTIONS

MADE BY-Aditya JanardhananGrade-XI-ARoll No.-8

Conic Sections - Definition• A conic section is a

curve formed by intersecting cone with a plane.

• There are four types of Conic sections.

1.Circle,2.ellipse, 3.parabola,4.hyperbola.

Conic Sections - Four Types

PARABOLA

What’s in a Parabola• A parabola is the set of all points in a plane such

that each point in the set is equidistant from a line called the directrix and a fixed point called the focus.

(x, y)

Parabola• The Standard Form of a Parabola that opens to the right

and has a vertex at (0,0) is……

)0(42 aaxy

Parabola• The Standard Form of a Parabola that opens to the left

and has a vertex at (0,0) is……

)0(42 aaxy

Parabola• The Standard Form of a Parabola that opens up and

has a vertex at (0,0) is……

ayx 42

Parabola

• The Standard Form of a Parabola that opens down and has a vertex at (0,0) is……

ayx 42

Parabola• The Standard Form of a Parabola

that opens to the left and has a vertex at (h,k) is……

)(4)( 2 hxaky

Parabola

• The Standard Form of a Parabola that opens up and has a vertex at (h,k) is……

)(4)( 2 kyahx

Parabola• The Standard Form of a Parabola that opens

down and has a vertex at (h,k) is……)(4)( 2 kyahx

Ellipse

What is in an Ellipse?• The set of all points in the plane, the sum of whose

distances from two fixed points, called the foci, is a constant. (“Foci” is the plural of “focus”, and is pronounced FOH-sigh.)

Ellipse

• The standard form of the ellipse with a center at (0,0) and a horizontal axis is……

12

2

2

2

by

ax

Ellipse• The ellipse with a center at (0,0) and a horizontal axis has

the following characteristics……

• Vertices ( a,0)• Co-Vertices (0, b)• Foci ( c,0)

1916

22

yx

Ellipse

• The standard form of the ellipse with a center at (0,0) and a vertical axis is……

12

2

2

2

ay

bx

Ellipse

• The ellipse with a center at (0,0) and a vertical axis has the following characteristics……

• Vertices (0, a) • Co-Vertices ( b,0) • Foci (0, c)

1819

22

yx

Ellipse

• The standard form of the ellipse with a center at (h,k) and a horizontal axis is……

1)()(2

2

2

2

bky

ahx

Ellipse• The ellipse with a center at (h,k) and a horizontal axis

has the following characteristics……

• Vertices (h a , k)• Co-Vertices (h, k b)• Foci (h c , k)

Ellipse

• The standard form of the ellipse with a center at (h,k) and a vertical axis is……

1)()(2

2

2

2

aky

bhx

Ellipse• The ellipse with a center at (h,k) and a vertical axis

has the following characteristics……

• Vertices (h, k a)• Co-Vertices (h b , k)• Foci (h, k c)

Hyperbola

The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

What is a Hyperbola?• The set of all points in the plane, the

difference of whose distances from two fixed points, called the foci, remains constant.

Hyperbola

• The standard form of the Hyperbola with a center at (0,0) and a horizontal axis is……

12

2

2

2

by

ax

Hyperbola

• The Hyperbola with a center at (0,0) and a horizontal axis has the following characteristics……

• Vertices ( a,0) • Foci ( c,0)

• Asymptotes: x

aby

Hyperbola

• The standard form of the Hyperbola with a center at (0,0) and a vertical axis is……

12

2

2

2

bx

ay

Hyperbola

• The Hyperbola with a center at (0,0) and a vertical axis has the following characteristics……

• Vertices (0, a) • Foci ( 0, c)

• Asymptotes: xbay

Hyperbola

• The standard form of the Hyperbola with a center at (h,k) and a horizontal axis is……

1)()(2

2

2

2

bky

ahx

Hyperbola• The Hyperbola with a center at (h,k) and a horizontal

axis has the following characteristics……

• Vertices (h a, k) • Foci (h c, k )

• Asymptotes:

)( hxabky

Hyperbola

• The standard form of the Hyperbola with a center at (h,k) and a vertical axis is……

1)()(2

2

2

2

bhx

aky

Hyperbola

• The Hyperbola with a center at (h,k) and a vertical axis has the following characteristics……

• Vertices (h, k a) • Foci (h, k c)

• Asymptotes: )( hxbaky

Circle

What is a circle ?

• A line forming a closed loop,every point on which is a fixed distance from a centre point.

Circle

• The Standard Form of a circle with a center at (0,0) and a radius, r, is…….. 222 ryx

center (0,0)radius = 2

Circles• The Standard Form of a circle with a center at (h,k) and

a radius, r, is……..

222 )()( rkyhx

    center (3,3)radius = 2