1. Topic: CONIC SECTION, POLAR COORDINATES AND SPACE GEOMETRY
CONIC SECTION
Ax2 + By2 + Dx + Ey + F = 0 is the general equation of conics
CIRCLE
Circle is set points which lie at a fixed distance called the Circle Tangent to Line Ax + By + C = 0
radius and from a fixed point called the center
Standard equation: r 2 = (x − h)2 + (y − k)2 | Ah + Bk + C |
Where: r=radius r =
(h, k) = center of the circle A 2 + B2
(x, y) = any point in the circle
General equation of a circle: x 2 + y 2 + Dx + Ey + F = 0
where D, E and F are arbitrary constants.
NOTE: r 2 < 0 no graph (imaginary circle)
r 2 = 0 graph is a single point
r 2 > 0 graph is a circle
PARABOLA
The locus of a moving point that its distance from a fixed point called the focus is equal to its distance from a fixed line called the
directrix.
Where: F = focus
v = vertex, midpoint of the segment
d1 = d2
e = eccentricity = 1
Standard equation: (y − k)2 = 4a (x − h) if the directrix is vertical
2
(x − h) = 4a (y − k) if the directrix is horizontal
Various opening of a parabola
General equation of a parabola: x2 + Dx + Ey + F = 0 is a parabola with horizontal directrix
y2 + Dx + Ey + F = 0 is a parabola with vertical directrix.
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2. Topic: CONIC SECTION, POLAR COORDINATES AND SPACE GEOMETRY
ELLIPSE
An ellipse is a set of points in a plane, such that the sum of the distances of each from two fixed point is a constant. The fixed points
are called the foci and the line through them is the axis.
Where: F1, F2 = focus
v1, v2 = vertex, midpoint of the segment
d1 + d2 = 2a
d c
e = eccentricity = 3 = < 1
d4 a
a
d=
e
a2 = b 2 + c 2
2b 2
LR = Latus Rectum =
a
(x − h)2 (y − k)2
Standard equation: 2
+ =1 if the major axis, M.A. is parallel to the x-axis
a b2
(y − k)2 (x − h)2
2
+ =1 if the major axis, M.A. is parallel to the y-axis
a b2
General equation of an ellipse: Ax 2 + By 2 + Dx + Ey + F = 0 where A and B is greater than 0
NOTE: M < 0 no graph (imaginary ellipse)
M = 0 graph is a single point
M > 0 graph is an ellipse
D2 E2
M = −F + +
4A 4B
HYPERBOLA
A hyperbola is a set of points in a plane, such that the difference of the distances of each from two fixed point is a constant. The
fixed points are called the foci and the line through them is the axis.
Where: F1, F2 = focus
v1, v2 = vertex, midpoint of the segment
d1 + d2 = 2a
d c
e = eccentricity = 3 = >1
d4 a
a
d=
e
c 2 =a2 +b 2
2b 2
LR = Latus Rectum =
a
Equation of Asymptote
Asymptote of a curve is a line when the perpendicular distance from a line to a curve approaches zero as the curve extends
indefinitely far from the origin.
y − k = ±m(x − h)
Where (h, k) is the center of the hyperbola and m is the slope, m=b/a if the axis is horizontal and m=a/b if the axis is
vertical. Use (+) for upward asymptote and (-) for downward asymptote.
(x − h)2 (y − k)2
Standard equation: − =1 if the transverse axis is parallel to the x-axis (Opens left and right)
a2 b2
(y − k)2 (x − h)2
2
− =1 if the transverse axis is parallel to the y-axis (Opens up and down)
a b2
Length of transverse axis (TA) = 2a
Length of conjugate axis (CA) = 2b
Length of the focal axis (FA) = 2c
General equation of hyperbola: Ax 2 + By 2 + Dx + Ey + F = 0 where A or B is less than 0
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3. Topic: CONIC SECTION, POLAR COORDINATES AND SPACE GEOMETRY
POLAR COORDINATES
The coordinates of a point in a plane are its distance from a Relationship between Rectangular and Polar
fixed point and its direction from a fixed line. Coordinates
Where: r = radius vector or modulus
θ = amplitude or argument or vectorial angle
x = r cos θ r = x2 + y2
y
y = r sin θ θ = tan −1
x
Curve Tracing on Polar Coordinates
1. The graph of r=k, where k is a constant and greater 4. The graph of r 2 = a2 sin 2θ or r 2 = a2 cos 2θ is a
than 0 (k>0) is a circle with center at the pole. lemniscate.
Illustration: Illustration:
r=2
r 2 = 4 sin 2θ
2. The graph of r = 2a sin θ , c(a,90o) and r=|a| or
r = 2a cos θ , c(a, 0o) and r=|a| is a circle that
passes through the pole.
Illustration:
r 2 = 3 cos 2θ
r = 4 sin θ
r = 4 cos θ
5. The graph of r = a sin(nθ) or r = a cos(nθ) is an n-
leaved rose
If n is an odd integer, then it is a n-leaved
rose
If n is an even integer, it is a 2n-leaved rose
If n = 1, then there is one petal and it is
circular
Illustration:
3. The graph of r = a(1 ± sin θ) or r = a(1 ± cos θ) is a r = 3 sin 4θ
cardiod.
Illustration:
r = 3(1 − cos θ)
r = 2 cos 2θ
r = 1 − cos θ
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4. Topic: CONIC SECTION, POLAR COORDINATES AND SPACE GEOMETRY
6. The graph of r = a ± b sin θ or r = a ± b cos θ is a limacon.
If |a| < |b| the graph is a limacon with inner loop
If |a| = |b| the graph is a cardiod
If |a| > |b| the graph is a limacon without a loop
Illustration:
r = 1 − 2 cos θ r = 4 + sin θ
SPACE GEOMETRY
Three Dimensional Coordinate System
Distance formula in three space Distance between a point (x, y, z)
and a plane
Ax 1 + By 1 + Cz 1 + D
d=
A 2 + B2 + C2
Midpoint formula
x + x2
x = 1
2
y1 + y 2
y =
2
z1 + z 2
z=
2
Cartesian coordinates (x, y z) d = (x 2 − x 1 )2 + (y 2 − y 1 )2 + (z 2 − z1 )2
Plane
The graph of Ax + By + Cz = D is a plane
Illustration: Cylinders
Any equation in two variables, represents a
Graph: 2x + y + 3z = 6 cylindrical surface, that is perpendicular to the two
variables and whose generating curve is the plane
curve whose equation is given.
Illustration:
Graph: x2+z2=4
Sphere:
Standard equation: (x − h)2 + (y − k)2 + (z − l)2 = r 2
General equation: x 2 + y 2 + z 2 + Dx + Ey + Fz + G = 0
Illustration:
Graph: x 2 + y 2 + z 2 − 2x + 4y + 6z − 2 = 0 Graph: y2=4x
C(1,-2,3) and r = 4
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5. Topic: CONIC SECTION, POLAR COORDINATES AND SPACE GEOMETRY
Quadratic Surfaces
1. Ellipsoid 3. Hyperboloid of Two Sheets
x2 y2 z2
2
+ 2
+ =1
a b c2
2. Hyperboloid of One Sheet x2 y2 z2
2
− 2
− =1
a b c2
4. Elliptic Paraboloid
x2 y2 z2
2
+ 2
− =1
a b c2
x2 y2
2
− =z
a b2
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