MATHEMATICS – CONIC SECTIONS

Slides:



Advertisements
Similar presentations
Lesson 10-1: Distance and Midpoint
Advertisements

Section 11.6 – Conic Sections
10.1 Conics and Calculus. Each conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone. CircleParabolaEllipse.
Conic Sections Parabola Ellipse Hyperbola
Conic Sections. (1) Circle A circle is formed when i.e. when the plane  is perpendicular to the axis of the cones.
Conic Sections Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Conic Sections Conic sections are plane figures formed.
Mathematics. Ellipse Session - 1 Session Objectives.
9.1 Conic Sections Conic sections – curves that result from the intersection of a right circular cone and a plane. Circle Ellipse Parabola Hyperbola.
Mathematics.
Review Day! Hyperbolas, Parabolas, and Conics. What conic is represented by this definition: The set of all points in a plane such that the difference.
Conics A conic section is a graph that results from the intersection of a plane and a double cone.
50 Miscellaneous Parabolas Hyperbolas Ellipses Circles
Conics can be formed by the intersection
Conic Sections Advanced Geometry Conic Sections Lesson 2.
Section 11.1 Section 11.2 Conic Sections The Parabola.
Conic Sections Curves with second degree Equations.
10.5 CONIC SECTIONS Spring 2010 Math 2644 Ayona Chatterjee.
Conic Sections.
Conic Sections There are 4 types of Conics which we will investigate: 1.Circles 2.Parabolas 3.Ellipses 4.Hyperbolas.
Conics, Parametric Equations, and Polar Coordinates Copyright © Cengage Learning. All rights reserved.
Distance The distance between any two points P and Q is written PQ. Find PQ if P is (9, 1) and Q is (2, -1)
Math Project Presentation Name Done by: Abdulrahman Ahmed Almansoori Mohammed Essa Suleiman Mohammed Saeed Ahmed Alali.
Conics. Conic Sections - Definition A conic section is a curve formed by intersecting cone with a plane There are four types of Conic sections.
INTRO TO CONIC SECTIONS. IT ALL DEPENDS ON HOW YOU SLICE IT! Start with a cone:
Conics A conic section is a graph that results from the intersection of a plane and a double cone.
10.1 Conics and Calculus.
CONIC SECTIONS.
An Ellipse is the set of all points P in a plane such that the sum of the distances from P and two fixed points, called the foci, is constant. 1. Write.
Chapter 12 THE PARABOLA 抛物线 5/7/2018 4:52:44 PM Parabola.
Objectives Identify and transform conic functions.
Conics A conic section is a graph that results from the intersection of a plane and a double cone.
Conics A conic section is a graph that results from the intersection of a plane and a double cone.
Analyzing Conic Sections
Chapter 6 Analytic Geometry. Chapter 6 Analytic Geometry.
10.1 Circles and Parabolas Conic Sections
Copyright © Cengage Learning. All rights reserved.
Conics Parabolas, Hyperbolas and Ellipses
6-3 Conic Sections: Ellipses
6.2 Equations of Circles +9+4 Completing the square when a=1
Conic Sections College Algebra
GRAPHS OF CONIC SECTIONS OR SECOND DEGREE CURVES
Worksheet Key 11/28/2018 9:51 AM 9.2: Parabolas.
12.5 Ellipses and Hyperbolas.
12.5 Ellipses and Hyperbolas.
Chapter 9 Conic Sections.
Section 10.2 – The Ellipse Ellipse – a set of points in a plane whose distances from two fixed points is a constant.
Ellipses Ellipse: set of all points in a plane such that the sum of the distances from two given points in a plane, called the foci, is constant. Sum.
Writing Equations of Conics
Review Circles: 1. Find the center and radius of the circle.
Unit 2: Day 6 Continue  .
Find the focus of the parabola
WORKSHEET KEY 12/9/2018 8:46 PM 9.5: Hyperbolas.
7.6 Conics
Chapter 6: Analytic Geometry
MATH 1330 Section 8.3.
MATH 1330 Section 8.3.
MATH 1330 Section 8.3.
Conic Sections: Hyperbolas
Analyzing the Parabola
MATH 1330 Section 8.3.
Analyzing Conic Sections
The Hyperbola Week 18.
THE ELLIPSE Week 17.
Section 11.6 – Conic Sections
What are Conic Sections?
Intro to Conic Sections
CONIC SECTIONS.
Conics Review.
Objectives & HW Students will be able to identify vertex, focus, directrix, axis of symmetry, opening and equations of a parabola. HW: p. 403: all.
Chapter 7 Analyzing Conic Sections
Presentation transcript:

MATHEMATICS – CONIC SECTIONS BY- ANANYA SEN 11-A

A conic section is the intersection of a plane and a cone.

By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola.

in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.

. CIRCLE A circle is the set of all points in a plane that are equidistant from a fixed point in a plane. *The circle is a special case of the ellipse

Circles in our surrounding-

EQUATIONS OF CIRCLE The equation of a circle with center at (a,b) and radius r units is (x−a)²+(y−b)² = r² If centre is (0,0) and radius is r, then equation of circle is x² + y² = r² If equation of circle is x^2+y^2+2ax+2by+c=0, then centre=(-a,-b) radius=√(a²+b²-c)

EXAMpLES- Q. Find the equation of the circle with-Centre(-2,3) and radius 4 A. a= -2, b= 3 and r= 4 Since, the equation of motion is(x−a)^2+(y−b)^2=r^2 So, on putting the values we get, (x+2)^2+(y-3)^2=r^2 Q. Find the centre and radius of the circle: (x^2+y^2-4x-8y-45=0) A. On comparing the given eq. with x^2+y^2+2ax+2by+c=0 Where, centre=(-a,-b) and radius=√(a^2+b^2-c) 2a= - 4→a= - 2, 2b= - 8 →b= - 4, c= -45 Centre = (2,4) and radius= √(4+16+45)= √65 Q. Find the equation of the circle whose centre lies on the line 4x+y=16 and which passes through the points(4,1) and (6,5). Let the eq. of the circle be, x^2+y^2+2gx+2fy+c=0 the circle passing through the points (4,1) and (6,5) 16+1+8g+2f+c=0 → 8g+2f+c= -17 ---(1)and 36+25+12g+10f+c=0 →12g+10f+c= -17----(2) On subtracting we get, g+2f= -11 ----------------(3) Since the centre (-g,-f) lie on the line 4x+y=16 ----------------(4) On subtracting eq. (3) and (4) we get, g= -3 On putting the value of g in eq. (3) we get, f = -4 Putting values of g and f in eq.(1) we get c=15 Therefore eq. of circle + x^2+y^2-6x-8y+15=0

PARABOLA A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.

Parabola in our surrounding-

Terms related to a parabola Axis The straight line passing through the focus and perpendicular to the directrix is called the axis of the parabola. Focus: The focus of a parabola is a fixed point in the interior of the parabola. Focal distance: The distance of a point on the parabola from its focus is called the focal distance of the point.

Vertex: The vertex of a parabola is the point where the parabola crosses its axis. When the coefficient of the x2 term is positive, then the vertex is the lowest point on the graph but in case it is negative the vertex will be the highest point on the graph. Directrix: A line perpendicular to the axis of symmetry is called the directrix. Latus Rectum: The latus rectum of a conic section is the chord through a focus parallel to the conic section directrix. The quantity 4a is known as the latus rectum. Half the latus rectum is called the semilatus rectum.

Equations of parabola

EXAMPLES- Q. Find the coordinates of the focus and the vertex, the eq. of the diretrix and the axis, and length of the latus rectum of the parabola y^2=8x. A. Comparing with y^2= -4ax, a=2 Focus=(-a,0)=(-2,0) Axis=x axis Eq. of Directrix, x-a=o so, x-2=0 Latus rectum=4a=8 Q. Find the coordinates of the focus and the vertex, the eq. of the diretrix and the axis, and length of the latus rectum of the parabola X^2= -16y A.On comparing with x^2= -4ay, a=4 Focus=(0,-a)=(0,-4) Axis=y axis Eq. of Directrix, y=a so, y=4 Latus rectum=4a=16 Q. Find the eq. of the parabola with vertex at the origin, passing through the point P(3,-4) and symmetric about the y-axis. Let its eq. be x^2= -4ay X=3 and y= -4 So, a=9/16 So, the required eq. is , x^2= (-9/4)y→ 4x^2+9y=0

Fig… ELLIPSE It is a path traced by a point which moves in a plane in such a way that the sum of its distance from two points in the plane is constant.

Ellipse in our surrounding-

Terms related to ellipse - The fixed points in the plane are the two focus points of an ellipse, jointly called the foci. The line segment passing through the foci of an ellipse and touching it is called the major axis of the ellipse The two points where the major axis touches the ellipse are called the vertices of the ellipse. The mid-point of the major axis, or the midpoint of the line segment joining the foci of an ellipse, is called the centre of the ellipse. ELLIPSE…

The line segment passing through the centre of an ellipse and perpendicular to the major axis is called the minor axis of the ellipse. Semi-major axis = ½ major axis = a Semi-minor axis = ½ minor axis = b Eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to onr=e of the foci and to one of the vertices of the ellipse. E=c/a, b^2=a^2(1-e^2). e<1

Equations of ellipse (0,0) (0,0) Focus- (±ae,0) Vertices-(±a,0) Directrix- x = ±(a/e) Latus rectum= 2b^2/a Distances between the foci= 2ae Distance between directrices- 2a/e Focus- (0,±ae) Vertices-(0,±a) Directrix- x = ±(a/e) Latus rectum= 2b^2/a Distances between the foci= 2ae Distance between directrices- 2a/e

Examples- Q. Find the lenghts of the major axes; coordinates of the vertices and the foci, the eccentricity and the length of the latus rectum of the ellipse x^2/4+y^2/25=1 A. Comparing with x^2/b^2+y^2/a^2=1 a^2=4, a=2 and b^2=25, b=5 b^2=a^2(1-e^2), e= √ 21/5 Therefore, foci= (0,± ae) ,(±√ 21 ) Vertices= (0,± a) ,=(0,±5) and (0, ±2) Major axis=2a ,10 Minor axis=2b ,4 Latus rectum= 2^2/a=8/5 Q. Find the equation of the ellipse, the ends of whose major axis are(± 3,0) and the ends of whose minor axis are(0, ± 2). Comparing with x^2/a^2+y^2/b^2=1 Its vertices are (±a,0), therefore, a=3 and lenth of the minor axis is 4.so,b=2 Hence, eq. = x^2/9+y^2/4=1

Q. Find the lengths of the major axes; coordinates of the vertices and the foci, the eccentricity and the length of the latus rectum of the ellipse x^2/36+y^2/16=1 Comparing with x^2/a^2+y^2/b^2=1 a^2=36, a=6 and b^2=16, b=4 b^2=a^2(1-e^2), e= √5/3 Therefore, foci= (± ae,0) ,(± 2 √5,0) Vertices= (± a,0) ,=(± 6,0) and (0, ± 4) Major axis=2a ,12 Minor axis=2b ,8 Latus rectum= 2b^2/a=13/3

HYPERBOLA It is the set of all the points in a plane, the difference of whose distance from the two fixed points in the plane is a constant.

HYPERBOLA IN SURROUNDINGS

TERMS RELATED TO HYPERBOLA The fixed points in the plane are the two focus points of an ellipse, jointly called the foci. The line segment passing through the foci of an ellipse and touching it is called the major axis of the ellipse.

Semi-major axis = ½ major axis = a Semi-minor axis = ½ minor axis = b The two points where the major axis touches the ellipse are called the vertices of the ellipse. The mid-point of the major axis, or the midpoint of the line segment joining the foci of an ellipse, is called the centre of the ellipse. The line segment passing through the centre of an ellipse and perpendicular to the major axis is called the minor axis of the ellipse. Semi-major axis = ½ major axis = a Semi-minor axis = ½ minor axis = b Eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to onr=e of the foci and to one of the vertices of the ellipse. E=c/a, b^2=a^2(e^2-1), e>1

EQUATION OF HYPERBOLA EQ… Focus- (±ae,0) Focus- (0,±ae) Vertices-(±a,0) Directrix- x = ±(a/e) Latus rectum= 2b^2/a Distances between the foci= 2ae Distance between directrices- 2a/e Focus- (0,±ae) Vertices-(0,±a) Directrix- x = ±(a/e) Latus rectum= 2b^2/a Distances between the foci= 2ae Distance between directrices- 2a/e

Examples- Q. Find the lengths of the transverse axis and conjugate axis; coordinates of the vertices and the foci, the eccentricity and the length of the latus rectum of the hyperbola x^2/16-y^2/9=1 A. Comparing with x^2/a^2-y^2/b^2=1 we get, A^2=16, a=4 and b^2=9, b=3 Therefore a=4,b=3 and e=5/4 {b^2=a^2(e^2-1)} Length of the transverse=2a=8 Length of the conjugate=2b=6 Vertices= (-a,o), (-4,0) and (a,0), (4,0) Foci=(± ae,0), (± 5,0) Latus rectum=2b^2/a=9/2 Q.Find the equation of the hyperbola whose vertices are(0, ±3)and the foci are (0, ± 5). A.Let the eq. be, y^2/a^2-x^2/b^2=1 Vertices is (0.±a) so, a=3 Foci is(0, ±ae) so e=5/3 By using {b^2=a^2(e^2-1)}, b=8 so, the eq. is y^2/36-x^2/64=1

A. . Comparing with y^2/a^2-x^2/b^2=1 we get, A^2=4 a=2 and b^2=9, b=3 Q. Find the lengths of the transverse axis and conjugate axis; coordinates of the vertices and the foci, the eccentricity and the length of the latus rectum of the hyperbola, y^2/4-x^2/9=1 A. . Comparing with y^2/a^2-x^2/b^2=1 we get, A^2=4 a=2 and b^2=9, b=3 Therefore a=2,b=3 and e=√13/2 {b^2=a^2(e^2-1)} Length of the transverse=2a=4 Length of the conjugate=2b=6 Vertices= (0,-a), (0-2) and (0,a), (0,2) Foci=(0,±ae), (0,± √13) Latus rectum=2b^2/a=9

APPLICATION BASED QUESTIONS ON CONIC SECTIONS- TRY THEM!!! Q.A water jet coming out of the small opening o of a fountain reaches its maximum height of 4m at a distance of 0.5m from the vertical. Find the height of the jet above the horizontal ox at a distance of 0.75m from the point o. Q. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100m long is suspended by vertical wires attached to the cable, the longest wire being 30m and the shortest being 6m. Find the length of a supporting wire at the vertex is the lowest point of the cable. Q. An arch is in the form of a semi-ellipse. It is 8m wide and 2m high at the centre. Find the height of the arch at a point 1.5m from one end. Q. A rod of length 12cm moves with its ends always touching the coordinate axes. Determine the equation of the path of a moving point p on the rod which is 3cm from the end in contact with the x- axis.

THANK YOU