Analytic geometry formulas (maths) PDF

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Analytic geometry...

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  Line segment

Line forms

A line segment P1P2 can be represented in parametric form by

Slope - intercept form:

y = mx + b

x = x1 + ( x 2 − x 1 )t

Two point form:

y − y1 =

y2 − y1 ( x − x1 ) x2 − x1

y = y1 + ( y 2 − y1 ) t 0≤ t ≤ 1

Point slope form:

Two line segments P1P2 and P3P4 intersect if any only if the numbers

y − y 1 = m ( x − x1 ) Intercept form

x2 − x1 x −x s= 3 1 x2 − x1 x3 − x 4

x y + = 1 ( a, b ≠ 0 ) a b Normal form:

x ⋅ cos σ + y sin σ = p

y 2 − y1 y 3 − y1 y2 − y1 y 3 − y4

x3 − x1 and

t=

y 3 − y1

x3 − x 4 y 3 − y 4 x2 − x1 y 2 − y1 x3 − x 4 y 3 − y 4

satisfy 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1

Parametric form:

x = x 1 + t cosα y = y1 + t sin α



Point direction form:



x −x1 y − y1 = A B

Area

where (A,B) is the direction of the line andP1 ( x1, y1) lies on the line. General form:

A ⋅ x + B ⋅ y + C = 0 A ≠ 0 or B ≠ 0

The area of the triangle formed by the three lines:

A1x + B1 y + C1 = 0 A2 x + B2 y + C2 = 0 A3 x + B3 y + C3 = 0 is given by

Distance The distance from Ax + By + C = 0 to P1 ( x1 , y1 ) is

d=

A ⋅ x1 + B ⋅ y1 + C

K= 2⋅

A2 + B 2

A1

B1 C 1

A2 A3

B2 C 2 B3 C 3

A1

B1

A2

B2

⋅

A2 A3

2

B 2 A3 B 3 ⋅ B3 A1 B1

The area of a triangle whose vertices are P1 ( x1 , y1 ) ,

Concurrent lines Three lines

A1x + B 1y + C 1 = 0 A2x + B 2y + C 2 = 0 A3x + B 3y + C 3 = 0 are concurrent if and only if:

A1

B1 C1

A2 B 2 C 2 = 0 A3

B3 C3

P2 ( x 2, y 2) and P3 ( x3 , y3 ) : x1 1 K = x2 2 x3

K=

y1 1 y2 1 y3 1

1 x 2 − x1 y 2 − y1 . 2 x3 − x1 y3 − y1

www.mathportal.org  Centroid



The centroid of a triangle whose vertices are P1 (x1, y 1) ,

P2( x 2, y 2) and P3 ( x3 , y3 ) :

Equation of a circle

 x + x + x y + y 2 + y3  ( x, y) =  1 2 3 , 1  3 3  

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that:

(x − a )2 + ( y − b )2

= r2

Circle is centred at the origin

Incenter

x2 + y 2 = r 2

The incenter of a triangle whose vertices areP1 (x1 , y1 ) ,

Parametric equations

x = a + rcos t y = b + r sin t

P2( x 2, y 2) and P3 ( x3 , y3 ) :  ax + bx2 + cx3 ay1 + by2 + cy3  (x , y ) =  1 ,  a +b +c  a +b +c 

where t is a parametric variable. In polar coordinates the equation of a circle is:

where a is the length of P2 P3 , b is the length of P1P3 , and c is the length of P1P2.

r2 − 2 rro cos ( θ − ϕ ) + ro2 = a2

Area A = r 2π

Circumference

Circumcenter The

circumcenter

of a triangle whose vertices are

c = π ⋅ d = 2π ⋅ r

P1(x 1, y 1) , P2 ( x2 , y2 ) and P3 (x3 , y3 ) :  x12 + y12 y1 1 x1 x12 + y12 1   2  2 y2 1 x2 x2 2 + y2 2 1   x2 + y2  x 2 +y 2 y 1 x x 2 +y 2 1  3 3 3 3 3 3  (x , y ) =  , x1 y 1 1 x1 y 1 1     2 x2 y 2 1   2 x2 y 2 1  x3 y3 1 x3 y3 1   

Theoremes: (Chord theorem) The chord theorem states that if two chords, CD and EF, intersect at G, then:

CD ⋅ DG = EG ⋅ FG (Tangent-secant theorem) If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then 2

DC = DG ⋅ DE

Orthocenter The

orthocenter

of a triangle whose vertices are

P1(x 1, y 1) , P2 ( x2 , y 2 ) and P3 (x3 , y 3 ) :  y1 x2 x3 + y 1 x + y2 y3  1 x + y 3 y1  y 2 x 3x1 + y  y x x +y 1 x + y 1y 2 3 1 2 (x , y ) =  , x1 y 1 1 x1 y 1   2 x2 y 2  2 x2 y 2 1  x3 y 3 1 x3 y 3  2 1 2 2 2 3

2 1 2 2 2 3

(Secant - secant theorem) If two secants, DG and DE, also cut the circle at H and F respectively, then:

DH ⋅ DG = DF ⋅ DE x1 1   x2 1  x3 1   1   1  1  

(Tangent chord property) The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord.

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Eccentricity:

The Parabola The set of all points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are always equal.

y 2 = 2px

Area: K = π ⋅ a⋅ b

2

The Hyperbola

Tangent line 2

Tangent line in a pointD (x0 , y0 ) of a parabola y = 2 px

y 0y = p (x + x 0 ) Tangent line with a given slope (m)

p 2m

Tangent lines from a given point Take a fixed pointP (x 0, y 0) .The equations of the tangent lines are

y − y 0 = m 1 ( x − x 0 ) and y − y 0 = m 2 ( x − x 0 ) where m1 = m1 =

Foci:

if a < b => F1 (0, − b2 − a2 ) F2 (0, b2 − a2 )

Parametric equations of the parabola:

y = mx +

a 2 − b2 a

if a > b => F1 (− a 2 − b2 , 0) F2 ( a2 − b2 , 0)

The standard formula of a parabola:

x = 2 pt y = 2 pt

e=

y0 +

y0 2 − 2 px0 and 2 x0

y0 −

y0 2 − 2 px0 2 x0

The Ellipse The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant.

The standard formula of a ellipse

x2 y 2 + =1 a2 b2 Parametric equations of the ellipse x = a cos t y = b sin t Tangent line in a point D( x0 , y0 ) of a ellipse:

x0 x y0 y + 2 =1 a2 b

The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

The standard formula of a hyperbola: 2

2

x y − 2 =1 2 a b Parametric equations of the Hyperbola

a sin t b sin t y= cost x=

Tangent line in a point D (x 0 , y 0 ) of a hyperbola:

x0 x y 0 y − 2 =1 a2 b

Foci: if a > b => F1 (− a 2 + b2 , 0) F2 ( a2 + b2 , 0) if a < b => F1 (0, − b2 + a2 ) F2 (0, b2 + a2 )

Asymptotes:

b b x and y = − x a a a a if a < b => y = x and y = − x b b if a > b => y =

www.mathportal.org  Plane forms Point direction form: x − x1 y − y1 z − z1 = = a b c where P1(x1,y1,z1) lies in the plane, and the direction (a,b,c) is normal to the plane.

General form:

Ax + By +Cz + D = 0 where direction (A,B,C) is normal to the plane.

Intercept form: x y z + + =1 a b c

Equation of a plane The equation of a plane through P1(x1,y1 ,z1) and parallel to directions (a1,b1,c1) and (a2,b2,c2) has equation

x − x1

y − y1

z − z1

a1

b1

c1

a2

b2

c2

The equation of a plane through P1(x1,y1,z1 ) and P2(x2,y2,z2), and parallel to direction (a,b,c), has equation

x − x1 x2 − x1

y − y1 y 2 − y1

a

b

z − z1 z 2 − z1 = 0 c

Distance

this plane passes through the points (a,0,0), (0,b,0), and (0,0,c).

x− x3

y − y3

x1 − x3 x2 − x3

y1 − y3 y2 − y3

The distance of P1(x1,y1,z1) from the plane Ax + By + Cz + D = 0 is

Ax1 + By1 + Cz 1

d=

Three point form z − z3 z1 − z3 = 0 z2 − z3

Normal form:

x cos α + y cos β + z cos γ = p Parametric form:

x = x1 + a1s + a 2t y = y 1 + b1s + b 2t z = z1 + c1 s + c2 t where the directions (a1,b1,c1) and (a2,b2,c2) are parallel to the plane.

Angle between two planes:

A2 + B 2 + C 2

Intersection The intersection of two planes

A1x + B1y + C 1z + D1 = 0, A2 x + B2 y + C2 z + D2 = 0, is the line

x − x1 y − y1 z − z1 , = = a b c where

B1

C1

B2

C2

C1

A1

C2

A2

A1 A2

B1 B2

a=

b=

c=

The angle between two planes:

A1x + B1 y + C1z + D1 = 0 A2x + B 2 y + C 2 z + D2 = 0 is

arccos

=0

b x1 =

A1 A2 + B1B2 + C1C 2 A1 2 + B1 2 + C1 2 A2 2 + B 2 2 + C 2 2

The planes are parallel if and only if

A1 B1 C1 = = A2 B 2 C 2 The planes are perpendicular if and only if

A1A2 + B1B 2 +C 1C 2 = 0

y1 =

D1 D2

C1 D −c 1 C2 D2

B1 B2

a 2 + b2 + c 2 D A1 D1 C1 c 1 −c D2 A2 D2 C 2

a 2 +b 2 + c 2 D B1 D1 a 1 −b D2 B2 D2 z1 = 2 2 a +b +c2

A1 A2

If a = b = c = 0, then the planes are parallel....