conic sectioins
TRANSCRIPT
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