Pre Cal Unit 8
foci
(plural of focus) - central points used to generate a conic section
Find the standard equation for the ellipse. x^2 − 2x + 1 + 3y^2 − 12y + 12 = 9
(x − 1)2/9 + (y − 2)2/3= 1
Find the standard-form equation for the parabola.vertex: (2, 2); directrix: y = 0
(x − 2) 2 = 8 (y − 2)
Write the standard equation for the ellipse.center: (2, 1); vertices: (5, 1) and (−1, 1); minor axis length of 4
(x − 2)^2/3^2 + (y − 1)^2/2^2 = 1
Write the standard equation for the ellipse. x^2 − 6x + 9 + 2y^2 − 4y + 2 = 4
(x − 3)^2/4 + (y − 1)^2/2 = 1
Write the standard equation for the circle.center: (h,k)=(4,7) radius: 1
(x − 4)^2 + (y − 7)^2 = 12
Write the standard equation for the circle.center: (h, k) = (5,9) radius: 3
(x − 5)^2 + (y − 9)^2 = 3^2
Write the standard equation for the ellipse.foci: (0, 3), (0, −3); major axis length of 10
(x)^2/4^2 + (y)^2/5^2 = 1
Write the standard equation for the ellipse. center: (3,2); vertices: (−1,2) and (7,2); foci: (0,2),(6,2)
(x−3)^2/16 + (y−2)^2/7 = 1
Write the standard equation for the ellipse.center: (3,2);(3,2); vertex: (7,2);(7,2); minor axis length of 6
(x−3)^2/4^2 + (y−2)^2/3^2 = 1
Convert to standard equation for the ellipse. x^2 − 16y + 4y^2 − 8x = −16
(x−4)^2/4^2 + (y − 2)^2/2^2=1
Find the standard-form equation for the parabola.vertex: (3, 1); directrix: x=4
(y − 1)^2 = −4 (x − 3)
Determine whether the equation represents a circle, parabola, ellipse, or hyperbola. Justify your answer by the three rules.
-2x^2 + 5x - 3y^2 + 7y = 0 __parabola__ Rule 1: x^2 and y^2 are multiplied by the different numbers with the different sign Rule 2: A= −1, B= 2, C= −1; B^2 − 4AC < 0; 2^2 − 4(−1)(−1) < 0; −16 < 0 Rule 3: AC > 0; −1(−3) > 1; 3>1
The eccentricity of a circle equals ____.
0
Find the eccentricity (4 decimal places). Planet X-38 in the Lost Galaxy orbits its star at a closest distance of 1.9 AU. The size of the major axis of its elliptical orbit is about 5.4 AU. Its star is one focus of its elliptical path 0.0072 AU from center.
0.0027
Find eccentricity: Assume the distance from the Sun to the planet Uranus at the closest is 18.4 AU. The size of the major axis of its elliptical orbit is 40.2 AU. The Sun is at focus of the planet's orbit at a point 0.9487 AU from orbital center.
0.0472
Find the eccentricity (4 decimal places). Uranus orbits the Sun in an elliptical orbit with a major axis of 40.2 AU. The Sun is at one focus 0.9487 AU from the center of the planet's elliptical orbit
0.0472
Find eccentricity: Assume Jupiter orbits the Sun at a distance from 4.95AU (closest average) to 5.46 AU (furthest average). The major axis of its elliptical path is 10.92AU. The Sun sits at a focus point of 0.2649 AU from the center of the planet's elliptical orbit.
0.0485
Find eccentricity: The major axis of Saturn's elliptical orbit is 20.24AU. the Sun is at focus 0.5485 AU from the center of the orbit.
0.0542
Find where the filament of a light bulb should be placed in a flashlight whose parabolic mirror has a diameter of 5 centimeters and depth of 2 centimeters. (Find answer to 4 decimal places.)
0.7813 cm
Eccentricity of a hyperbola is always greater than _____________________.
1
How far from the vertex should the receiver be placed in a parabolic-shaped satellite dish with a diameter of 8 feet and depth of 2.5 feet?
1.6 feet
Arella launched a toy rocket from the ground. The rocket's initial velocity was 100 fps. How high was the rocket 5 seconds after launch?
100 feet
Find the furthest distance from the Sun. (to the nearest tenth) Pluto has an orbital eccentricity of 0.2488. The Sun is at a focus of its elliptical path at 9.8276 AU.
39.5
Determine whether the equation represents a circle, parabola, ellipse, or hyperbola. Write the answer on the blank. Justify your answer by the three rules.
3x^2 + 3(y^2−y) + 6 = 0 __parabola__ Rule 1: Both squared variables are multiplied by the same positive number. Rule 2: A = −1, B = 2, C = −1; B^2 − 4AC < 0; −16<0 Rule 3: AC>0; 3(3)>0; 9>0
Henry fired an arrow straight up into the air from a deck that was 6 ft high. The arrow left the bow with a velocity of 180 feet per second (fps). the equation for finding the height of a projectile is S = −16t^2 + vt + k . How high was the arrow after 7 seconds of flight?
482 feet
A daredevil dives off a 25-foot cliff falling along a parabolic arc into the water. How far from the base of the cliff will she enter the water? The equation for the arc is y = −x^2 + 25.
5 feet from the base
The equation for the height of a projectile launched vertically and following a parabolic path is: S=−16t2+vt+k.S=-16t2+vt+k. ( vv is velocity and kk is height from the ground) A carbine rifle is fired vertically from a platform. The end of the rifle is 12 feet from the ground. The muzzle velocity (speed of the bullet leaving the rifle) is 1,990 feet per second (fps). How high off the ground will the bullet be 3 seconds after firing?
5,838 feet
Find the eccentricity of the hyperbola. The length of the transverse axis is 8;8; length of conjugate is 6
5/4
Find the eccentricity of the hyperbola.Length of transverse axis is 8; length of conjugate axis is 6
5/4
Determine whether the equation represents a circle, parabola, ellipse, or hyperbola. Write the answer on the blank. Justify your answer by the three rules.
7x^2 - 3 + 4y = 0 __parabola__ Rule 1: only one variable is squared Rule 2: A = 7, B = 0, C = 0; B^2 - 4AC = 0; 0-0 < 0 = 0 Rule 3: AC = 0; 7(0) = 0; 0 = 0
Reflecting telescopes use curved mirrors to gather and focus light from object millions or billions of miles away. The Hubble Telescope has a primary mirror 94.5 inches in diameter. If the depth of the mirror is 59.5 inches, where would its focus be located? Find the answer to two decimal places.
9.38 inches
A reflecting telescope primary mirror is 100 inches in diameter. Where would the focus of the mirror be located if its depth is 63 inches? (Answer to the nearest hundredth.)
9.92 inches
Find the eccentricity of the hyperbola. The length of the conjugate is 8√2; distance between the vertices is 14
9/7
Find the eccentricity of the hyperbola. distance between vertices is 14; distance between foci is 18
9/7
The measure of the distance from the Earth to the Sun is one __________________.
AU
Write the general second-degree equation for conics.
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Change the equation to standard form if needed. Find the center ___ and ___ radius . x^2+ y^2 + 4y − 6x − 16 = 0
Center: (3, -2) Radius: √29
Change the equation to standard form. Find the center and radius . x^2 + y^2 + 2y − 8x − 4 = 0
Center: (4, -1) Radius: √21
___________________ is the measure of the ovalness of an ellipse.
Eccentricity
Use the indicated rule to determine the type of conic from the equation.
Rule 1: b. Only one squared variable (x^2 or y^2) Type: parabola
Use the indicated rule to determine the type of conic from the equation.
Rule 1: x^2 and y^2 are multiplied by different numbers with the same sign Type: ellipse
locus; loci (plural)
a collection of points that satisfy a condition and form a curve
orbit
a curved path around a central object
directrix
a fixed line used to generate a curve
focus
a given point that is as far from a point on a parabola as the directrix
paraboloid
a hollow parabolic shape created by rotating a parabola about its axis
degenerate
a limited and less functional form
tangent
a line that touches a curve at a single point
eccentricity
a measure of the ovalness of an ellipse
vertex
a point farthest from the center of an ellipse
Find a,b,(h,k), and write the standard equation for the hyperbola . foci (−3,3),(−3,−1) and vertices (−3,2),(−3,0)
a: 1 b: √3 (h, k): (-3, 1) hyperbola: (y + 3)^2/1 − (x − 1)^2/ (√3)^2 = 1
. Find a, b, (h,k), and write the standard equation for the hyperbola . vertices : (1,3),(1,−3); Asymptotes; y=±(x−1)y=±(x-1) CAREFUL! Note the ratio in the asymptote equation for a vertical transverse axis.
a: 1 b: √3 (h, k): (1, 0) hyperbola: . y^2/9 − (x − 1)^2/1 = 1
Find a____, b____, (h,k)_______, and write the standard equation for the hyperbola ______.vertices (1, 3), (5, 3); asymptotes: y = 1/2x + 1 1/2; y = − 1/2x + 4 1/2
a: 2 b: 1 (h, k): (3, 3) hyperbola: (x − 3)^2/4 − (y − 3)^2/1 = 1
Find a____ b____ (h, k)____ and write the standard equation for the hyperbola .vertices (2, 0), (2, 4); foci (2, −2), (2, 6)
a: 2 b: √12 (h, k): (3, 3) hyperbola: (y − 2)^2/4 − (x − 2)^2/12 = 1
Find a____,b___, (h,k),___ and write the standard equation for the hyperbola . Foci (0, 1), (8, 1) and vertices (2, 1), (6, 1)
a: 2 b: √3 (h, k): (4, 1) hyperbola: (x − 4)^2/4 − (y − 1)^2/12 = 1
Find a____, b___, (h,k)___ and write the standard equation for the hyperbola_____ .vertices: (3,1),(−3,1); foci: (4,1),(−4,1)
a: 3 b: √ (h,k): (0, 1) hyperbola: x^2/9 − (y − 1)^2/7 = 1
Find a, b, (h,k) and write the standard equation for the hyperbola . foci (3, 0), (3, −6) and asymptotes: y = − x; y = x − 6
a: 3 b: √8 (h, k): (3, -3) hyperbola: (y + 3)^2/9 − (x − 3)^2/9 = 1
Find a, b, (h,k), and write the standard equation for the hyperbola . vertices (1,4),(5,4); asymptotes y = 5/2x − 3 1/2y = -5/2x + 11 1/2
a: 5 b: √3 (h, k): (3, 4) hyperbola: (x − 4)^2/9 − (y − 2)^2/7 = 1
Eccentricity of a hyperbola is greater than 1 because _____________.
c>a
Fill in the blanks: (x+5)^2/12 + y^2/16 = 1
center: (-5,0) vertices: (-5,4) (-5,-4) foci: (-5,2) (-5,-2)
Change the equation to standard form if needed. Find the center ____ and_____ radius . (x − 4)^2 + (y − 3)^2 = 36
center: (4,3) radius: √36 = 6
Fill in the blanks: (x − 2)^2/4^2 + (y − 3)^2/2^2 = 1
center: (h, k) = (2, 3) vertices: (h + a, k) and (h − a, k )= (6, 3) and (−2, 3) c^2 = a^2 − b^2 foci: (h + c, k) and (h − c, k) = (2+23√,3) and (2−23√,3)
Fill in the blanks: (x − 0)^2/9 + (y − 1)^2/25 = 1
center: = (0, 1) vertices: (h, k + a) and (h, k − a) = (0, 6) and (0, −4) c^2 = a^2 − b^2 foci: (h,k+c) and (h,k−c) = (0, 5) and (0, −3)
Complete the blanks given: x^2/1 + y^2/7 = 1
center:= (0,0) vertices:= (0,√7) and (0,√-7) foci:= (0, √6) and (0, √-6)
Fill in the blanks: x^2/4^2 + y^2/1^2 = 1
center:= (0,0) vertices:= (4, 0) and (−4, 0) foci:= (√15, 0) and (−√15 ,0)
Fill in the blanks: (x − 1)^2/5^2 + (y − 5)^2/4^2 = 1
center:= (1, 5) vertices:= (6, 5) and (−4, 5) foci:= (4, 5) and (-2, 5)
If the foci of an ellipse lie at the center point it becomes a(n) ________________.
circle
The axis drawn perpendicular to the transverse axis is called the _____________ axis.
conjugate
The sum of the distances of points on an ellipse from the foci is _________________ .
constant
A fixed line used to form a curve is called the ______________.
directrix
radius
distance from the center of a circle to a point on the circle
Determine the type of conic from the equation: 7x^2 − 2x + 3y + 5y^2 = 0
ellipse
Orbits of the planets are __________.
elliptical
Two fixed points used to generate an ellipse are called ___________.
foci
Incoming rays parallel to the axis are directed to the _____________ in a parabolic reflector.
focus
One central point used to generate an ellipse is called a(n) ________________.
focus
One of two fixed points used to generate an ellipse is called ___________.
focus
The ____________ is a point that is as far from any point on a parabola as the directrix.
focus
Determine the type of conic from the equation: 3x^2 − 4x − y^2 + 1 = 0
hyperbola
The _____________ is a collection of points that form a curve.
locus
The _______________ axis is the longer axis in an ellipse.
major
The length of the _______________ axis is equal to the sum of the distances of a point on an ellipse from the foci.
major
conjugate
one of a pair joined together, as in two axes
Determine the type of conic from the equation: x^2 − y + 5 = 0
parabola
Name the basic conics.
parabola ellipse hyperbola circle
Rays coming from the focus are reflected ______________ to the axis.
parallel
The conjugate axis is ____ to the transverse axis.
perpendicular
Name the degenerate conics.
point two intersecting lines line
transverse
relating to the axis that passes through the foci of a hyperbola
Use the indicated rule to determine the type of conic from the equation.
rule 1 : x^2 and y^2 are multiplied by numbers with different signs type: ellipse
A ______________ is a line that touches a curve at a single point.
tangent
asymptote
the line a curve approaches as the curve goes to infinity
vertex
the lowest or highest point where the line of a parabola turns
equidistant
the same distance
apex
the vertex or tip
The ______________axis passes through the foci of a hyperbola.
transverse
The axis that passed through the foci of a hyperbola is called the ____________ axis.
transverse
A point farthest from the center of an ellipse is a __________.
vertex
Convert to the standard form and find the vertex, directrix, and focus. x^2 − 2x + 1 = 4y − 16
vertex (1, 4) directrix y = 3 focus (1, 5)
Convert to the standard form and find the vertex, directrix, and focus. y^2 − 4x − 2y − 7 = 0
vertex (−2, 1) directrix x = −3 focus (−1, 1)
Convert to the standard form to find the vertex, directrix, and focus. Y^2 + 16 = 8y + 4x - 4
vertex: (1, 4) directrix: x = 0 focus: (2, 4)
Convert to the standard form to find the vertex, directrix, and focus. x^2 − 4x + 4 + 2y + 6 = 0
vertex: (2, −3) directrix: y=−2 1/2 focus: (2, −3 1/2)
Convert to the standard form and find the vertex, directrix, and focus. x^2 − 10x + 25 = 2y − 4
vertex: (h,k) = (5,2) directrix: y = 1 1/2 focus: (h,k+p) = (5, 2 1/2)
Convert to the standard form and find the vertex, directrix, and focus. y^2 − 2y + 1 − x = 3
vertex: x = −3 1/4 directrix: x = 1 1/2 focus: (h+p,k) = (−2 3/4, 1)
The two points at both ends of the major axis in an ellipse are called __________.
vertices
Determine whether the equation represents a circle, parabola, ellipse, or hyperbola. Write the answer on the blank. Justify your answer by the three rules.
x + y^2 + y − 8 = 0 __parabola__ Rule 1: only one variable is squared Rule 2: A = 0, B = 0, C = 1; B^2 − 4AC = 0; 0 − 0 = 0; 0 = 0 Rule 3: AC > 0; -1 (-3) > 1; 3 > 1
Write the standard equation for the circle.center: (h,k)=(0,-5) radius: 5
x^2 + (y + 5)^2 = 5^2
Write the standard equation for the circle.center: (h,k)= (0,0) radius: 16
x^2 + y^2 = 16
Find the standard-form equation for the parabola.vertex: (0, 2); focus: (0, 4)
x^2 = 8 (y − 2)
Find the standard-form equation for the parabola.focus: (0, 0); directrix: y = 4
x^2 = −8 (y − 2)
Convert to standard equation for the ellipse. 4x^2 + 9y^2 − 36 = 0
x^2/3^2 + y^2/2^2 = 1
Find the standard-form equation for the parabola.vertex: (-3,0); focus: (−1,0)
y^2 = 8 (x + 3)
Determine whether the equation represents a circle, parabola, ellipse, or hyperbola. Justify your answer by the three rules.
−2x^2+5x−3y^2+7y=0 __ellipse__ (circle, parabola, ellipse or hyperbola) Rule 1: ___x^2 and y^2 are multiplied by the different numbers with the different sign __ Rule 2: A=−2, B=0, C=−3; B^2 − 4AC <0; 0^2 − 4(−2)(−3) <0;−24 <0 Rule 3: AC>0; −2(−3)>0; 6>0
Find the eccentricity of the hyperbola. 8) Find the eccentricity of the hyperbola. vertices: (0, 2), (6, 2); asymptotes; y = 2 ± 2/3 (x − 3)
√13/3
Solve. Mercury's mean distance from the Sun is 0.387 AU. Find the time to the nearest tenth it will take the planet to complete one orbit.
≈ 0.24 years
It takes Mars 1.88 Earth-years (687 days) to complete one orbit of the Sun. What is its average distance from the Sun? (two decimal places) Use the table to approximate your answer.
≈ 1.52 AU
Find the time to orbit the Sun once: Asteroid Eros orbits the Sun from 1.783 AU to 1.133AU. How long does it take for it to orbit once around the Sun? (2 decimal places)
≈ 1.760 years
Find the average distance of the orbit from the Sun to the nearest whole number. Use the Cube Root table. (Round to the nearest tenth) The asteroid Vesta, orbits the Sun approximately every 3.6 years.
≈ 2.4 AU
Find the average distance of the orbit from the Sun to the nearest whole number. Use the Cube Root table. (Round to the nearest tenth) The minor planet Ceres orbits the Sun once every 4.6 years
≈ 2.8AU
The average orbital distance from the Sun for comet Chiron is 8.5 AU. How long will it take it to travel once around the Sun? (Answer to the nearest hundredth.)
≈ 24.78 years
Find the time to orbit the Sun once: The mean distance of Uranus from the sun is 19.2 AU. How long does it take the planet to complete one orbit around the Sun? (nearest whole number)
≈ 84 years
