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Analytic Geometry Formulas Fo rmulas 1. Lines in two dimensions dimensions Line segment
Line forms
A line segment P1 P2 can be represented in parametric
Slope - intercept i ntercept form:
form by
y = mx+ b
x = x1 + ( x2 − x1 ) t
Two point form:
y− y1
=
y2
− y1
x2
− x1
(
x− x1 )
0 ≤ t ≤ 1
Point slope form:
y − y1
=
y = y1 + ( y2 − y1 ) t
Two line segments P1P2 and P3 P4 intersect if any only if
m ( x − x1 )
the numbers
Intercept form
x
+
a
y
=1
b
( a, b ≠ 0 )
s=
Normal form:
x⋅ cos σ
ysin σ = p
+
x2 − x1
y2
− y1
x3 − x1
y3 − y1
x2 − x1
y2
x3 − x4
y3 − y4
and
− y1
t
=
x3
− x1
y3
− y1
x3
− x4
y3
− y4
x2
− x1
y2
− y1
x3
− x4
y3
− y4
1 satisfy 0 ≤ s ≤ 1 and 0 ≤ t ≤
Parametric form:
x = x1 + t cos α y = y1 + tsin α Point direction form:
x − x1
=
2. Triangles in two dimensions
y − y1
A
Area
B
where (A,B) is the direction of the line and P1 ( x1 , y1 ) lies on the line. General form:
A⋅ x + B ⋅ y + C = 0 A ≠ 0 o r B ≠ 0
The area of the triangle formed by the three lines:
A1 x + B1 y + C1 = 0 A2 x + B2 y + C2 = 0 A3 x + B3 y + C3 = 0 is given by
Distance 0 to P1 ( x1 , y1 ) is The distance from Ax+ By+ C = d =
A⋅ x1 + B ⋅ y1 + C A
2
+
B
K =
2⋅
2
2
A1
B1
C1
A2
B2
C2
A3
B3
C3
A1
B1 A2
B2
A2
B2 A3
B3
⋅
⋅
A3
B3
A1
B1
The area of a triangle whose vertices are P1 ( x1 , y1 ) ,
Concurrent lines
P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
Three lines
A1 x
+
B1 y
+
C1 = 0
A2 x
+
B2 y
+
C2
=
0
A3 x
+
B3 y
+
C3
=
0
are concurrent if and only if:
A1
B1
C1
A2
B2
C2 = 0
A3
B3
C3
K
=
K =
x1
y1
1
x2
y2
1
x3
y3
1
1 x2 − x1
y2
2 x3 − x1
y3 − y1
1 2
− y1
.
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Centroid
3. Circle
The centroid of a triangle whose vertices are P1 ( x1 , y1 ) ,
P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
x1 + x2 + x3 y1 + y2 , 3 3
( x, y ) =
Equation of a circle +
y3
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
The incenter of a triangle whose vertices are P1 ( x1 , y1 ) ,
+
y2
=
r2
Parametric equations
P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
x = a + r cos t
ax + bx2 + cx3 ay1 + by 2 + cy3 ( x, y ) = 1 , a +b +c a +b +c
y = b + rsin t 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 P1 P3 , and c is the length of PP 1 2.
r2
−
2rro cos (θ
)
− ϕ + ro
2
=
a2
Area A = r 2π
Circumference
Circumcenter The
circumcenter of a triangle whose vertices are P1 ( x1 , y1 ) , P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
x12 + y12 y1 1 x1 x12 + y12 1 2 2 2 2 1 1 x y y x x y + + 2 2 2 2 2 2 x2 + y2 y 1 x x2 + y2 1 3 3 3 3 3 3 ( x, y ) = , x1 y1 1 x1 y1 1 2 x2 y2 1 2 x2 y2 1 x3 y3 1 x3 y3 1
c
=
π ⋅d
(Chord theorem) The chord theorem states that if two chords, CD and EF, intersect at G, then:
CD ⋅ DG
orthocenter of a triangle whose vertices are P1 ( x1 , y1 ) , P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
y1 x2 x3 + y12 1 x12 + y2 y3 2 2 y2 x3 x1 + y2 1 x2 + y3 y1 y xx +y2 1 x2+y y 3 1 2 3 3 1 2 ( x, y ) = , x1 y1 1 x1 y1 2 x2 y2 2 x2 y2 1 x3 y3 1 x3 y3
x1
1
x2 1 x3 1 1 1 1
=
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
The
2π ⋅ r
Theoremes:
DC Orthocenter
=
=
DG⋅ DE
(Secant - secant theorem) If two secants, DG and DE, also cut the circle at H and F respectively, then:
DH ⋅ DG = DF ⋅ DE (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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4. Conic Sections
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.
a 2 − b2
e=
a
Foci: if a > b => F1 ( − a
The standard formula of a parabola: 2
y = 2 px
2
−b
if a < b => F1 (0, − b
Parametric equations of the parabola:
2
2
2 2 , 0) F2 ( a − b ,0)
−a
2
2 ) F2 (0, b
−a
2
)
Area:
2
= π ⋅ a ⋅ b
K
x= 2 pt y= 2 pt
The Hyperbola
Tangent line Tangent line in a point D( x0 , y0 ) of a parabola
2
y = 2 px
y0 y = p ( x + x0 ) Tangent line with a given slope (m)
y= mx+
x
2
2m
a
2
Take a fixed point P( x0 , y0 ) .The equations of the tangent lines are
y− y0
=
m1 ( x− x0 ) and
y− y0
=
m2 ( x− x0 ) where
m1
2
y+
y
0
=
−2
0
px 0
2 x0 2
y−
y
0
=
0
−2
and
px 0
The standard formula of a ellipse a2
+
b2
=1
x = acos t y = bsin t Tangent line in a point D( x0 , y0 ) of a ellipse:
a2
+
2
=1
Parametric equations of the Hyperbola x =
a sin t b sin t cos t
x0 x
y0 y
a
b
− 2
2
=1
Foci: if a > b => F1 ( − a
y0 y b2
2
+b
=1
2
2
2 2 , 0) F2 ( a +b , 0)
+a
2
2 ) F2 (0, b
Asymptotes: if a > b => y
=
2
Parametric equations of the ellipse
x0 x
b
2
if a < b => F1 (0, − b
The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant.
y
y
Tangent line in a point D( x0 , y0 ) of a hyperbola:
2 x0
2
−
y =
The Ellipse
x
The standard formula of a hyperbola:
p
Tangent lines from a given point
m1
The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
if a < b => y =
b a a b
x and y = − x and y = −
b a a b
x x
+a
2
)
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5. Planes in three dimensions Plane forms
The equation of a plane through P1(x1,y1,z1) and parallel to directions (a 1,b1,c1) and (a2,b2,c2) has equation
Point direction form: x − x1
y − y1
=
a
=
Equation of a plane
z − z1
x − x1
y − y1
z − z1
c
a1
b1
c1
a2
b2
c2
b
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.
y
+
a
z
+
b
x − x1
y − y1
z − z1
x2
y2
z2
x − x3
y − y3
z − z3
x1
−
x3
y1
−
y3
z1
− z3 =
x2
−
x3
y2
−
y3
z2
− z3
The distance of P1(x1,y1,z1) from the plane Ax + By + Cz + D = 0 is
+
x − x1
Angle between two planes:
a
=
b
=
c=
The angle between two planes:
A1 x + B1 y + C1 z + D1
=
A2 x + B2 y + C2 z + D2 A1 A2 B1
A2
=
B1
=
B2
=
+
+
x1 B1 B2
C1
2
C 1
B2
C 2
C1
A1
C2
A2
A1
B1
A2
B2
0 +
A2
2
=
+
B2
2
+
c
2
C2
y1
=
C1
C1 C2 = 0
z1
=
C1
D2
C2
=
0,
=
z − z1
0,
,
D1
B1
D2
B2
2
A1
D2
A2
−c
2
D1
C1
D2
C2
2
+b +c
D1
B1
D2
B2 a2
−c
+b +c
D1 a2
a
C2 +
=
c
D1 a2
C1C2
The planes are perpendicular if and only if
A1 A2 + B1 B2
y − y1
B1
b
The planes are parallel if and only if
A1
C2
b
0
is
+
+
where
2
=
a
where the directions (a1,b1,c1) and (a2,b2,c2) are parallel to the plane.
A1
B2
+
is the line
zcos γ = p
a2 t
= z1 + c1 s + c2 t
2
A2
A1 x + B1 y + C1 z + D1
y = y1 + b1 s + b2 t
arccos
Ax1 + By1 + Cz1
The intersection of two planes
Parametric form:
z
c
A2 x + B2 y + C2 z + D2
ycos β
+ a1 s +
0
Intersection 0
Normal form:
x = x1
− z1 =
b
d =
Three point form
+
− y1
Distance
this plane passes through the points (a,0,0), (0,b,0), and (0,0,c).
xcos α
− x1
a
=1
c
0
The equation of a plane through P1(x1,y1,z1) and P2(x2,y2,z2), and parallel to direction (a,b,c), has equation
Intercept form: x
=
−b 2
2
D1
A1
D2
A2
+b +c
2
If a = b = c = 0, then the planes are parallel.
