P.o.D. – Given the equation 9𝑥 2 −
𝑦 2 + 54𝑥 + 10𝑦 + 55 = 0
1.) Find the center.
2.) Find the foci.
3.) Find the vertices.
4.) Find the equations of the
asymptotes.
1.) C(-3,5)
2.) 𝐹(−3 ±
3.) 𝑉 (
−10
3
√10
, 5)
3
−8
, 5) , 𝑉 (
3
, 5)
4.) 𝑦 − 5 = ±3(𝑥 + 3)
10.6 – Parametric Equations
Learning Target(s) – I can
evaluate sets of parametric
equations for given values of the
parameter; sketch parametric
equations; rewrite parametric
equations in rectangular form;
find sets of parametric
equations for graphs.
Vector Equation of a Line:
⟨𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 ⟩ = 𝑡⟨𝑎1 , 𝑎2 ⟩
EX: Write a vector equation
describing a line passing
through P(3,-9) and parallel to
𝑎⃗ = ⟨−1,2⟩
⟨𝑥 − 3, 𝑦 + 9⟩ = 𝑡⟨−1,2⟩
Parametric Equation of a Line:
Given 𝑃1 (𝑥1 , 𝑦1 ) and vector 𝑎⃗ =
⟨𝑎1 , 𝑎2 ⟩, the parametric equation
𝑥 = 𝑥1 + 𝑡𝑎1
is 𝑦 = 𝑦 + 𝑡𝑎 .
1
2
EX: Find the parametric
equations of a line parallel to 𝑏⃗⃗ =
⟨3, −5⟩ and passing through the
point at (4,3).
𝑥 = 4 + 3𝑡
𝑦 = 3 − 5𝑡
*Show how to graph on the
calculator.
EX: Write the parametric
1
equations of 𝑦 = 𝑥 + 7.
2
x and t are independent
variables.
1
𝑥 = 𝑡, 𝑦 = 𝑡 + 7
2
EX: In the national air race at
Reno, Nevada, a pilot passes the
starting gate at a speed of 411
mph, and 20 seconds later a 2nd
pilot flies by at 413.3 mph. They
are racing for the next 500 miles.
Use parametric equations to
model the situation. Assume
that both planes maintain a
constant speed.
a. How long is it until the 2nd
plane overtakes the first?
1st Plane: x=411t
2nd Plane: x=413.3(t-.0056)
20seconds=0.0056hr
411𝑡 = 413.3(𝑡 − .0056) →
411𝑡 = 413.3𝑡 − 2.31448 →
−2.3𝑡 = −2.31448 →
−2.31448
𝑡=
= 1.0063 ℎ𝑜𝑢𝑟𝑠
−2.3
b.) How far have both planes
traveled when the 2nd plane
overtakes the first?
x=411(1.0063)=413.588
x=413.3(1.0063-.0056)=413.589
EX: Write an equation in slopeintercept form of the line whose
parametric equations are x=-2+t
and y=4-3t.
Solve both equations for t.
𝑥 = −2 + 𝑡
𝑥+2=𝑡
𝑦 = 4 − 3𝑡
3𝑡 = 4 − 𝑦
4−𝑦
𝑡=
3
Set the two equations equal and
solve for y.
𝑥+2 4−𝑦
=
1
3
3𝑥 + 6 = 4 − 𝑦
𝑦 = −3𝑥 − 2
Do the following on your own:
a.) Write a vector equation of
the line through 𝑃1 (5, −3)
and is parallel to 𝑎⃗ = ⟨1,2⟩.
b.) Write parametric equations
for the line parallel to 𝑎⃗ =
⟨2, −2⟩ and passing through
the point (1,3).
c.) Write the parametric
equations of y=2x-5.
d.) Write an equation in slopeintercept form of the line
whose parametric
equations are x=-4+2t and
y=5-t.
a.) ⟨𝑥 − 5, 𝑦 + 3⟩ = 𝑡⟨1,2⟩
b.) x=1+2t, y=3-2t
c.) x=t, y=2t-5
1
d.) 𝑦 = − 𝑥 + 3
2
Modeling Motion Using
Parametric Equations.
Projectile Motion:
⃗⃗⃗⃗𝑥 | = |𝑉
⃗⃗ | cos 𝜃
|𝑉
⃗⃗⃗⃗
⃗⃗
|𝑉
𝑦 | = |𝑉 | sin 𝜃
EX: Find the initial horizontal
velocity and vertical velocity of
a stone kicked with an initial
velocity of 18 ft/s at an angle of
37 degrees with the ground.
⃗⃗⃗⃗𝑥 | = 18 cos 37° =
Horizontal: |𝑉
14.375 𝑓𝑡/𝑠
⃗⃗⃗⃗
Vertical: |𝑉
𝑦 | = 18 sin 37° =
10.833 𝑓𝑡/𝑠
Parametric Equations for the
Path of Projectile Motion:
⃗⃗ | cos 𝜃
𝑥 = 𝑡|𝑉
1 2
⃗⃗ | sin 𝜃 − 𝑔𝑡 ,
𝑦 = 𝑡|𝑉
2
Where g is acceleration due to
gravity.
𝑚
𝑓𝑡
𝑔 = 9.8 2 𝑜𝑟 𝑔 = 32 2
𝑠
𝑠
EX: Sammy Baugh of the
Washington Redskins has the
record for the highest average
punting record for a lifetime
average of 45.16 yards. Suppose
that he kicked the ball with an
initial velocity of 26 yards per
seconds at an angle of 72
degrees.
a.) How far has the ball
traveled horizontally and
what is its vertical height
at the end of 3 sec?
𝑥 = 3(26) cos 72° ≈ 24.103 𝑦𝑑𝑠
1
𝑦 = 3(78) sin 72° − (32)(3)2
2
≈ 78.547 𝑓𝑡 𝑜𝑟 26.182 𝑦𝑑𝑠.
b.) Suppose that the kick
returner lets the ball hit
the ground instead of
catching it. What is the
hang time?
1
0 = 𝑡(78) sin 72° − (32)𝑡 2
2
0 = 74.1824𝑡 − 16𝑡 2
16𝑡 2 − 74.1824𝑡 = 0
𝑡(16𝑡 − 74.1824) = 0
𝑡=0
16𝑡 − 74.1824 = 0
16𝑡 = 74.1824
𝑡 = 4.6364
EX: Suppose that a softball
pitcher throws a ball at an angle
of 5.1 degrees with the
horizontal at a speed of 85 mph.
The distance of the pitcher’s
mound to home plate is 60.5 ft. If
the pitcher releases the ball 2.9
feet above the ground, how far
above the ground is the ball
when it crosses home plate?
⃗⃗ | cos 𝜃 = 𝑡(85𝑚𝑝ℎ) cos 5.1°
𝑥 = 𝑡|𝑉
85𝑚𝑖 5280𝑓𝑡
1ℎ𝑟
1𝑚𝑖𝑛
×
×
×
1ℎ𝑟
1𝑚𝑖
60𝑚𝑖𝑛 60𝑠𝑒𝑐
= 124.67 𝑓𝑡/𝑠
𝑥 = 124.67𝑡 cos 5.1°
1 2
⃗⃗ | sin 𝜃 − 𝑔𝑡 + ℎ
𝑦 = 𝑡|𝑉
2
1
𝑦 = 𝑡(124.67) sin 5.1° − (32)𝑡 2 + 2.9
2
Distance to home plate: 60.5 ft.
60.5 = 124.67𝑡 cos 5.1°
0.4872 = 𝑡
Substitute.
𝑦 = .4872(124.67) sin 5.1°
1
− (32)(.4872)2 + 2.9
2
= 4.5015 𝑓𝑡 𝑎𝑏𝑜𝑣𝑒 ℎ𝑜𝑚𝑒 𝑝𝑙𝑎𝑡𝑒.
EX: Sketch the curve
represented by the parametric
equations x=t and y=-2t, -2<t<2.
EX: Sketch the curve
represented by 𝑥 = 2 cos 𝜃 and 𝑦 =
2 sin 𝜃, 0 ≤ 𝜃 ≤ 2𝜋
Upon completion of this lesson,
you should be able to:
1. Write the vector equation of
a line.
2. Write the parametric
equation of a line.
3. Write parametric equations
of projectile motion.
For more information, visit
http://tutorial.math.lamar.edu/Classes/CalcI
I/ParametricEqn.aspx
HW Pg.776 3-42 3rds, 69-76