LECTURE 3: Analytic Geometry
Example: Directed Distance Concept
Show that the following are the vertices of a
parallelogram. A(2,0) B(5,‐1) C(11,4) and D(8,5)
Example: Distance Between Two Points
Show that the following are the vertices of a right
triangle and find its area. A(‐1,‐2) B(2, 1) C(‐3,6)
Example: Distance Between Two Points
What are the coordinates of a point 3 units from
the y axis and at a distance of √5 from (5,3)
(3,2) (3,4)
Example: Distance Between Two Points
Find the coordinates of a point equidistant from
P(1,‐6) Q(5,‐6) R(6,‐1)
(3,-3)
Example: Area by Coordinates
Find the area of the quadrilateral ABCD
A(1,‐6) B(5,‐6) C(6,‐1) D(‐2,6)
(3,-3)
Example: Area by Coordinates
Three consecutive vertices of a parallelogram are
located at A(1,6), B(10,9) and C(3,‐5). Determine
a. Coordinates of the 4th vertex
b. Its area
(3,-3)
Example: Subdivision Concept
The line segment with end points A(‐1,‐6) and
B(3,0) is extended beyond point A to a point C so
that C is 4 times as far from B as from A. Find the
coordinates of point C.
(3,-3)
Example: Subdivision Concept
Three vertices of a parallelogram are at
(5, 12) (2,7) and (6,1). Determine all
possible coordinates of the 4th vertex.
(3,-3)
Example: Subdivision Concept
Find the coordinates of the point of
intersection of the medians of the
triangle with vertices A(‐3,‐7) B(3,1)
C(‐8,2)
C(‐8, 2)
B (3, 1)
A (‐3, ‐7)
(3,-3)
Example: Subdivision Concept
In what ratio does the point (‐1,8) divide
the line segment from (‐3,10) to (4,3)?
A (‐3, 10)
B(‐1, 8)
C (4, 3)
(3,-3)
Example: Slope and Inclination
Do the following points lie on a straight
line? (‐1,8) (‐3,10) (4,3)?
(‐3, 10)
(‐1, 8)
4, 3)
(3,-3)
Example: Slope and Inclination
Show that the following points are the
consecutive vertices of a parallelogram
and determine a pair of adjacent
angles. Compute its area.
A(‐3,3) B(‐2,‐2) C(8,3) D(7,8)
A (‐3,3)
D (7,8))
C (8, 3)
B (‐2, ‐2)
(3,-3)
Example: Slope and Inclination
Show that the following points are the
vertices of a right triangle and
determine its two acute angles.
Compute its area.
A(‐1,6) B(‐3, 4) C(2,‐1)
B (‐3,4)
A (‐1,6)
C (2,‐1)
(3,-3)
Example: The Straight Line
A triangle has its vertices at A(‐8,2),
B( 2, 7) and C( 6, ‐4)
B (2,7)
a. Side AC
b. Perpendicular bisector to side AB
c. Altitude to AC
d. Median to the shortest side
e. A line with a slope of 2/3 and
through the midpoint of AB
A (‐8,2)
Find the equation of the following:
C (6,‐4)
(3,-3)
Example: The Straight Line
Find the equation of a line through
(‐2, 6) with x intercept half the y
intercept. What is its inclination?
x y
+ =1
a b
(0,b)
(‐2,6)
b
(a,0) a
(3,-3)
Example: The Straight Line
a. Derive the perpendicular distance
of a point from a line
Ax
y+
B
+
0
=
C
x2 ‐ x1
(x1,y1)
d
(x2,y2)
θ
(3,-3)
Example: The Straight Line
A line passes through (3,2) and forms
with the axes a triangle with an area
of 12 sq units.
a. What is the equation of this line?
b. What is the distance of the origin
from this line?
(0,b)
b
x y
+ =1
a b
(3,2)
a
(a,0)
(3,-3)
Example: The Straight Line
Find the equations of the line parallel
to 3x + 4y – 12 = 0 and passing at a
distance of ±3 from the given line.
required line
3
3x + 4y – 12 = 0
0,3
4,0
(3,-3)
Example: The Straight Line
If the inclination (θ) of the line (k+1)x
+ ky – 3 = 0 is arctan(‐2), find k.
θ
(3,-3)
Example: The Straight Line
Find the equation of the family of lines
passing through the intersection of 2x
– 5y = 10 and 3x + 2y = 12. What is the
equation of a specific member of this
family which
a. pass through (‐2, 4)
b. y intercept of 6
c. parallel to 3x – 4y – 12 = 0
d. bisect the angle between the two
given lines
(3,-3)
Example: The Straight Line
Concurrency of Three Lines
L1: A1x + B1y + C1 = O
L2: A2x + B2y + C2 = O
L3: A3x + B3y + C3 = O
A1 B1
A2 B2
A 3 B3
C1
C2 = 0
C3
CIRCLE
- The set of all points equidistant from a fixed point
Example: Circle Satisfying
3 Conditions
Determine the equation of the circle
whose radius is 5, center on the line
x = 2 and tangent to the line
3x – 4y + 11 = 0
x=2
3x – 4y + 11 = 0
r=
5
(2,k)
Example: Center – Radius Form
(Standard Equation)
Determine the equation of the circle with
center at (2,‐5) and passing through (3,4).
C(3, 4)
r
O(2,‐5)
Example: Circle Satisfying
3 Conditions
Determine the equation of the circle
passing through (‐3,6), (‐5,2) and (3,‐6)
A(‐3,6)
r
B(‐5,2)
r
O(h,k)
r
C(3,‐6)
Example: Circle Satisfying
3 Conditions
Determine the equation of the circle
tangent to 3x + y + 14 = 0 and x + 3y +
10 = 0 and radius of √10.
L 1:3x
+3
y+
10
=
=0
√1
0
4
+y+1
0
O(h,k)
r=√
10
r=
L2 :x
Tangential Distance
‐it is the length of the tangent segment from the circle to an
external point (x1,y1)
Formulas:
1. When circle is written in standard
form ( center radius form )
dT = ( x1 − h) + ( y1 − k ) − r
2
2
2
2. When circle is written in
general form
2
2
dT = x1 + y1 + ax1 + by1 + c
dT
(h.k)
(x1,y1)
Example: Tangential Distance
1. Find the tangential distance from
(‐7,‐2) to the circle (x + 1)2 + (y ‐ 2)2 =
26.
2. Find the tangential distance from
(0,6) to the circle x2 + y2 ‐ 2x + 2y – 23
=0
Example: Radical Axis and
Common Chord
Radical axis
x
y
Line of centers
Note: line of centers is perpendicular to the common tangent/radical axis
Line segment xy is called geodesic – shortest distance between two circles
Example: Radical Axis and
Common Chord
Common Chord
Example: Common Chord,
Radical Axis, Line of Centers
Determine the equation of the common
chord/radical axis and line of centers of
the circles x2 + y2 + 2x ‐ 4y – 4 = 0 and
x2 + y2 ‐ 6x + 2y – 6 = 0
Parabola
- The set of all points such that its distance from a
fixed point (Focus) equals its distance from a fixed line
(Directrix)
- It is the locus (graph) of all points equidistant from
a fixed point (Focus) and a fixed line (Directrix)
Parabola
axis of parabola
latus rectum
(LR = 4a)
Focus (F)
a
V
a
directrix
Equations of Parabola
1. general format
(one variable)2 = ± 4a(other variable)
2. Standard Forms
a. (x‐h)2 = 4a(y‐k) vertex at (h,k) opens up
b. (x‐h)2 = ‐4a(y‐k) vertex at (h,k) opens down
c. (y‐k)2 = 4a(x‐h) vertex at (h,k) opens to the right
d. (y‐k)2 = ‐4a(x‐h) vertex at (h,k) opens to the left
3. General Equations
a. y = ax2 + bx +c
when a > 0 opens up
when a < 0 opens down
b. x = ay2 + by +c when a > 0 opens to the right
when a < 0 opens to the left
Example: Identifying the Parts of a Parabola
Given its Equation
Sketch and identify the parts and attributes
of y2 ‐ 8x + 6y + 17 = 0
Example: Equation Determination
Determine the equation of parabola with
vertex at (5,‐2) and focus at (5, ‐4)
Example: Equation Determination
A parabola has its axis parallel to x‐axis and
one end of its latus rectum at (‐7/2,‐3) and
vertex at (‐3, ‐2). Find its equation.
Example: Equation Determination
Determine the equation of parabola with
vertex on the line y = 2x, axis parallel to the x
axis and passes through (3/2,1) and (3,‐4)
Example: Application Problems
A window has the shape of a parabola with
vertex at the top and axis vertical. It is 2
meters wide at the base and 4 meters high.
How wide is it halfway?
Example: Application Problems
A horizontal pipe weighing 40 kg/m is
supported by vertical hangers equally
spaced as shown and these hangers are
supported by a cable supported at A and
B.
a. How far from A is the lowest point?
b. How far below B is a point on the cable
which is 40 m horizontally away?
Ellipse
- The set of all points the sum of whose distances
from two fixed points is a constant.
Attributes/Parts of an Ellipse
minor axis
Vertex 1
Focus 1
center
Focus 2
Vertex 2
2b2
LR =
a
major axis
distance between foci = 2c
major diameter = 2a
the two directrices
distance = 2a/e
eccentricity (e) = c/a
0<e<1
minor diameter = 2b
latus rectum
Equations of Ellipse
1. Standard forms
(with coordinates of center at (h,k))
(x ‐ h)2 (y ‐ k)2
+
= 1 → major axis // to x‐axis
2
2
a
b
(y ‐ k)2 (x ‐ h)2
+
= 1 → major axis // to y‐axis
2
2
a
b
2. General Equation
(A ≠ B and A,B > 0)
Ax2 + By2 + Cx + Dy + E = 0
Note: Apply Factoring to Reduce the General Equation
to Standard Form
Example: Determining the Parts and
Sketching the Ellipse
Identify the parts, attributes and
determine the area of
16x2 + 9y2 ‐ 64x + 54y + 1 = 0
Example: Determining the Equation
and Parts of Ellipse
Find the equation of the ellipse
with vertices at (‐3,‐2) and (1,‐2)
and which passes through (‐2,‐1).
5x2 + 9y2 = 180
Example: Determining the Equation
and Parts of Ellipse
Find the equation of the locus of
a point which moves so that its
distance from (4,0) is equal to
two thirds of its distance from
the line x = 9.
5x2 + 9y2 = 180
Example: Determining the
Eccentricity of Ellipse
If the length of the latus rectum
of an ellipse is 3/4 of the length
of its minor axis, find its
eccentricity.
HYPERBOLA
-The set of all points the difference of whose distances
from two fixed points is a constant.
Attributes/Parts of Hyperbola
Conjugate axis
c
F1
V1
C (h,k)
b
a V2
F2
latus rectum
LR = 2b2/a
Axis of hyperbola
Asymptotes
ideal rectangle
eccentricity (e) = c/a
e>1
Example: Determining the
Equation of Hyperbola
The foci of hyperbola are (4,3)
and (4,‐9) and the length of the
conjugate axis is 4√5,
a. Find its eccentricity.
b. Equation of the hyperbola.
c. Equations of its asymptotes.
Example: Determining the
Graph and Attributes
Given the curve: 9x2 – 2y2 = 18
a. Determine the eccentricity.
b. Coordinates of center.
c. Coordinates of vertices.
d. Length of latus rectum
e. Equations of directrices
g. Angle between the two
asymptotes
Example: Determining the
Graph and Attributes
Given the curve: 64x2‐100y2‐
1024x + 1000y – 4804 = 0
a. Determine the eccentricity.
b. Coordinates of center.
c. Coordinates of vertices.
d. Length of latus rectum
e. Equations of directrices
Example: Conjugate Hyperbolas
Write the equation of the hyperbola
conjugate to the hyperbola 4x2 – 3y2
+ 32x + 18y + 25 = 0.
Tangents To Conics
Case 1. Point of Tangency is given
Case 2. Direction of Tangent is given
Case 3. An External Point is Given
Procedure:
a. Let the equation of tangent be
1. y = mx + b ( for case 2)
2. y ‐ y1 = m( x ‐ x1 ) ( for case 1 and 3 )
b. Substitute y in the given equation of conic
and write the resulting equation in the form
Ax2 + Bx + C = 0
c. Equate the discriminant to zero and solve for
b (case 2) or m (case 1 and case 3)
d. Substitute back b or m to step a and the
resulting equation is the equation of tangent.
Example: Equation of Tangent
Find the equation of tangent to
y2 = 8x at the point (2, 4).
Example: Equation of Tangent
Find the coordinate of a point on x2
= 16y at which there is a tangent of
slope ½.
Example: Equation of Tangent
Find the equations of the tangent
and normal to 2x2 – y2 + 7 = 0 if
tangent passes through (5,‐1)
Polar Coordinates
Transformation of Rectangular
Coordinates to Polar Coordinates
Equations of Transformation
1. x 2 + y2 = r2
PR (x , y)
PP (r , θ)
r
y
θ
x
2. x = rcosθ
3. y = rsinθ
y
4.
= tanθ
x
Example: Polar Coordinates
Given the two points A(‐4, 5π/4) and
B(3, ‐4).
a. Convert each to rectangular or polar
form
b. Compute the distance between the
two points using polar coordinates
Example: Polar Coordinates
Convert the equation x2 + y2 – 6y = 0
to polar form
Example: Polar Coordinates
Convert the equation r = 6sinθ to
rectangular form
Example: Polar Coordinates
Sketch the graph of r2 = 9sin2θ