WWW.IDREES.PK

Chapter – 6

Conic Sections Introduction to Conic Sections: CONIC SECTIONS Conic Section: a section (or slice) through a cone

Did you know that by taking different slices through a cone you can create a circle, an ellipse, a parabola or a hyperbola?

Cones

Circle

Ellipse

Parabola

Hyperbola

C

Eccentricity: Eccentricity: how much a Conic Section (a circle, ellipse, parabola or hyperbola) varies from being circular?

Different values of eccentricity make different curves: • • • • •

For eccentricity = 0 we get a Circle For 0 < eccentricity < 1 we get an Ellipse For eccentricity = 1 we get a Parabola For eccentricity > 1 we get a Hyperbola For infinite eccentricity we get a Line

1

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

WWW.IDREES.PK Circle: A set of points such that distance of each point from a fixed point (center) remains constant. The constant distance from the center is called radius of the circle. Equations of Circle:

Conditions

Equation

The standard form where center ( h, k ) and radius r

( x − h) + ( y − k )

If center ( 0,0 ) and radius r

x2 + y 2 = r 2

Parametric Equations of circle

x = r cos θ , y = r sin θ

2

If end points of the diameter of the circle A ( x1 , y1 ) and

B ( x2 , y2 ) .

2

= r2

( x − x1 )( x − x2 ) + ( y − y1 )( y − y2 ) = 0

General Equation of Circle: The general equation of the circle is given as

x 2 + y 2 + 2 gx + 2 fy + c = 0 Properties and Results from General Equation:

1. The general equation of the circle involves three constants i.e. g , f , c 2. The general equation of a circle is a second degree equation in which coefficients of x 2 and y 2 are 1. 3. The general equation does not contain the term involving the product xy .

(−g, − f ) Center of the Circle

Radius of the Circle

i.e.

 coefficient of x coefficient of y  ,− −  2 2   •

r = g 2 + f 2 − c > 0 , Real Circle

•

r = g 2 + f 2 − c = 0 , Point Circle

•

If r = g 2 + f 2 − c < 0 , Imaginary Circle.

Tangent and Normal: • Tangent: A straight line that touches the curve at a point without cutting the curve. • Normal: A straight line perpendicular to the curve at the point of tangency.

•

Equation of tangent to the circle x 2 + y 2 = r 2 at P ( x1 , y1 ) is xx1 + yy1 = r 2 .

2

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

WWW.IDREES.PK •

Equation of normal to the circle x 2 + y 2 = r 2 at P ( x1 , y1 ) is xy1 − yx1 = 0 .

•

Equation of tangent to the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 at P ( x1 , y1 ) is

xx1 + yy1 + g ( x + x1 ) + f ( y + y1 ) + c = 0 . •

Equation of normal to the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 at P ( x1 , y1 ) is

( x − x1 )( y1 + f ) − ( y − y1 )( x1 + g ) = 0 . Important Theorems: • The point P ( x1 , y1 ) lies outside, on or inside the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 according as x12 + y12 + 2 gx1 + 2 fy1 + c > 0 ,

• •

x12 + y12 + 2 gx1 + 2 fy1 + c = 0

x12 + y12 + 2 gx1 + 2 fy1 + c < 0 .

respectively. Two tangents can be drawn to a circle from any point outside the circle. Let P ( x1 , y1 ) be a point outside the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 , then length of either tangent drawn from P ( x1 , y1 ) to the circle =

•

or

x12 + y12 + 2 gx1 + 2 fy1 + c .

The line y = mx + c intersects the circle x 2 + y 2 = r 2 in at the most two points, the points are

1. Real and Distinct if, r 2 (1 + m 2 ) − c 2 > 0 2. Real and Coincident if, r 2 (1 + m 2 ) − c 2 = 0 3. Imaginary if, r 2 (1 + m 2 ) − c 2 < 0 •

Condition of Tangency: The line y = mx + c touches x 2 + y 2 = r 2 if c = ± r 1 + m 2 . Some Important Information about the Circle

3

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

WWW.IDREES.PK

Parabola: A set of points in a plane such that the distance of each point from a fixed point (focus, F) is equal to its distance from a fixed straight line (directrix, M).

Terms Related to Parabola: 1. Axis of Parabola: The line through focus and perpendicular to directrix is called axis of parabola. 2. Vertex: The midpoint of the perpendicular line joining focus and directrix is called vertex (or turning point) of parabola. 3. Focal Chord: A chord of the parabola through focus is called focal chord. 4. Latus Rectum: The focal chord perpendicular to the axis of the parabola is called latus rectum of the parabola. 5. Eccentricity: The ratio of the distance of any point on the parabola to its distance from directrix.

4

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

WWW.IDREES.PK Standard Forms of Parabola: y 2 = 4ax

y 2 = −4ax

x 2 = 4ay

x 2 = −4ay

Focus

( a, 0 )

( − a, 0 )

( 0, a )

( 0, −a )

Vertex

( 0,0 )

( 0, 0 )

( 0, 0 )

( 0, 0 )

4a

4a

4a

4a

x=a

x = −a

y=a

y = −a

x = −a

x=a

y = −a

y=a

y=0

y=0

x=0

x=0

1

1

1

1

Equation

Length of Latus Ractum Equation of Latus Rectum Equation of Directix Axis Eccentricity

Graph

Ellipse: A set of points in a plane such that the distance of each point from a fixed point (focus, F) bears a constant ratio (Eccentricity, 0 < e < 1 ) to its perpendicular distance from a fixed straight line (directrix, M).

5

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

WWW.IDREES.PK

Terms Related to Ellipse: 1. 2. 3. 4. 5. 6. 7.

Vertices: The points on the standard ellipse where it crosses the x-axis. Co–Vertices: The points on the standard ellipse where it crosses the y-axis. Center: The midpoint of the line joining vertices (or co-vertices). Major Axis: The line joining vertices is called major axis. Minor Axis: The line joining co-vertices is called minor axis. Latus Recta: The chords perpendicular to major axis passes through foci are called latus recta. Eccentricity: The ratio of the distance of any point on the ellipse from the focus to its distance from the directrix.

Standard Forms of Ellipse:

Equation

x2 y 2 + = 1, a > b a 2 b2

y2 x2 + = 1, a > b a 2 b2

Foci

( ± ae, 0 ) or ( ±c, 0 )

( 0, ± ae ) or ( 0, ±c )

Vertices

( ± a, 0 )

( 0, ±a )

Co Vertices

( 0, ±b )

( ± b, 0 )

2a

2a

y=0

x=0

2b

2b

Equation of Minor Axis

x=0

y=0

Length of Latus Rectum

2b 2 a

2b 2 a

Equation of Directrices

x=±

Length of Major Axis Equation of Major Axis Length of Minor Axis

Eccentricity

6

e=

a e

a2 − b2 , a>b a

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

y=± e=

a e

a 2 − b2 , a>b a

WWW.IDREES.PK

Graph

Hyperbola: A set of points in a plane such that the distance of each point from a fixed point (focus, F) bears a constant ratio (Eccentricity, e > 1 ) to its perpendicular distance from a fixed straight line (directrix, M).

7

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

WWW.IDREES.PK Terms Related to Hyperbola: 1. 2. 3. 4. 5. 6.

Vertices: The points on the standard hyperbola where it crosses the x-axis. Center: The midpoint of the line joining vertices (or co-vertices). Transverse Axis: The line joining vertices is called transverse axis. Conjugate Axis: The line joining co-vertices is called conjugate axis. Latus Recta: The chords perpendicular to major axis passes through foci are called latus recta. Eccentricity: The ratio of the distance of any point on the hyperbola from the focus to its distance from the directrix. 7. Asymptotes: In general, an asymptote is a line that approaches a curve but never touches. Every hyperbola has associated with two lines called asymptotes. Their point of intersection is center of the hyperbola. b a) Slope of asymptotes = ± , hyperbola opens side. a a b) Slope of asymptotes = ± , hyperbola opens up and down. b Standard Forms of Hyperbola:

x2 y 2 − =1 a 2 b2

y2 x2 − =1 a2 b2

( ± ae, 0 ) or ( ±c, 0 )

( 0, ± ae ) or ( 0, ±c )

Vertices

( ± a, 0 )

( 0, ± a )

Co Vertices

( 0, ±b )

( ±b, 0 )

2a

2a

y=0

x=0

2b

2b

Equation of Conjugate Axis

x=0

y=0

Length of Latus Rectum

2b 2 a

2b 2 a

Equation of Directrices

x=±

Equation Foci

Length of Transverse Axis Equation of Transverse Axis Length of Conjugate Axis

Eccentricity

8

e=

a e

a 2 + b2 a

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

y=± e=

a e

a 2 + b2 a

WWW.IDREES.PK

Graph

Tangent and Normal: 1. To parabola y 2 = 4ax is yy1 = 2a ( x + x1 ) .

x2 y2 xx yy 2. To ellipse 2 + 2 = 1 is 21 + 21 = 1 . a b a b 2 2 x y xx yy 3. To hyperbola 2 − 2 = 1 is 21 − 21 = 1 . a b a b −y 1. To parabola y 2 = 4ax is y − y1 = 1 ( x − x1 ) . 2a

Equation of tangent at point P ( x1 , y1 )

Equation of normal at point P ( x1 , y1 )

2. To ellipse

x2 y2 a 2 x b2 y is − = a 2 − b2 . + = 1 x1 y1 a2 b2

3. To hyperbola

x2 y 2 a2 x b2 y is + = a2 + b2 − = 1 x1 y1 a 2 b2

1. Parabola y 2 = 4ax if a = mc . Condition that a line y = mx + c is tangent to a cone

2. Ellipse

x2 y2 + 2 = 1 if c = ± a 2 m 2 + b 2 2 a b

3. Hyperbola

x2 y2 − = 1 if c = ± a 2 m 2 − b 2 a2 b2

Replace Generally to find an equation of tangent at a point P ( x1 , y1 ) , makes the replacements

x 2 by xx1 ,

in the given equations of the curve.

x by

y 2 by yy1 , xy by

xy1 + yx1 2

x + x1 y + y1 , y by 2 2

Identification of a Cone from the General Equation: The general equation of a cone ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0 represents: 1. A circle if h 2 − ab < 0 & a = b . 2. An ellipse if h 2 − ab < 0 & a ≠ b .

9

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])

WWW.IDREES.PK 3. A parabola if h 2 − ab = 0 . 4. A hyperbola if h 2 − ab > 0 .

a 5. A pair of straight lines, if h g

h b f

g f =0 c

Theorem:

If the axes are rotated about the origin through an angle θ ( 0 < θ < 90 ) , where θ is given by

tan 2θ =

10

2h , then the product term xy in general second degree vanished in the new coordinate axes. a −b

By Muhammad Idrees, Head of Mathematics Department, Islamia Boys College, Quetta, Pakistan. ([email protected])