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Worksheets3. Conic, equation of lines, (and planes), Distance btwn L,P TF
Total questions: 91
Worksheet time: 58mins
Find general form of the equation of a circle centered at (−3,5) passing through (1,−2).
22x^2 + 6y^2 - 10x - 31y = 0
22x^2 + 6y^2 - 10x - 31y = 0
22x^2 + 61063xy - 0 = 0
22x^2 + 6y^2 - 10x + 18y = 0
22x^2 + 61018xy - 0 = 0
Rewrite in standard form.
22x^2 + 4y^2 - 9x + 8y = 0
22(1)(2)xy + 94 = 0
22(1)(2)xy - 94 = 0
22(1)(2)xy - 49 = 0
22(2)(1)xy + 49 = 0
Given general form determine the y – intercept.
(0,-3)
(0,-2)
(-3,0)
(-2,0)
(1,0)
Determine the value of eccentricity for.
5/3
5/2
3/2
1
-1
Determine the foci coordinates of an ellipse.
(2, 3) ±
(3, 0) ±
(3, 0)
(2, 0) ±
(2, 3) ±
Rewrite the equation in standard form.
22x^2 + 4y^2 - 9x + 32y - 54 = 0
22(4)(3)xy + 94 = 0
22(4)(3)xy + 94 = 0
22(4)(5)xy - 0 = 0
22(4)(5)xy - 0 = 0
Identify the graph of the following equation as a parabola, circle, ellipse, straight line or hyperbola.
A circle
An ellipse
A straight line
A hyperbola
A parabola
Identify the graph of the following equation.
A hyperbola opening left and right
A hyperbola opening up and down
A parabola that opens right
A parabola that opens left
A parabola that opens up
Identify the graph of the following equation.
A parabola that opens right
A hyperbola opening up and down
A hyperbola that opens left and right
A parabola that opens left
A parabola that opens up
Find the x- and y-intercepts.
int: (1, 3, 5, 0); int: x-ercept y-ercepts none
int: (1, 2, 5, 0); int: (1, 0)
int: (15, 0); int: (1, 0)
int: (15, 0); int: (0, 1)
int: (1, 5, 0); int: (0, 2)
Rewrite in standard form.
22x^2 - 25y^2 + 64 - 200 = 0
22(4)(5)xy - 64 = 0
22(5)(5)xy - 64 = 0
22(4)(5)xy - 25 = 0
22(4)(5)xy - 2564 = 0
Find the right characteristics to the following ellipse.
Center: (-2; 0)
Foci: (-2; 6.3) and (-2; -6.3)
Vertices: (-2; 5) and (-2; -5)
Co-vertices: (2; 0) and (-7; 0)
Find the right characteristics to the following ellipse.
Center: (4; 3)
Foci: (-1.8; 3) and (-6.2; 3)
Vertices: (-1; -3) and (-7; -3)
Co-vertices: (-4; -1) and (-4; -5)
Find the right characteristics to the following ellipse.
Center: (7; 2)
Foci: (12.7; 2) and (1.3; 2)
Vertices: (13; -2) and (1; -2)
Co-vertices: (7; 0) and (7; -4)
Find the right characteristics to the following ellipse.
Center: (8; 6)
Foci: (8; 9) and (8; 2)
Vertices: (8; 10) and (8; 2)
Co-vertices: (10; -6) and (6; -6)
Find the right characteristics to the following ellipse.
Center: (-7; -1)
Foci: (-7; 2) and (-7; -4)
Vertices: (-7; 2) and (-7; -4)
Co-vertices: (-6; 1) and (-8; 1)
Find the right characteristics to the following ellipse.
Center: (-6; 2)
Foci: (10.9; 2) and (1.1; 2)
Vertices: (10; 2) and (0; 2)
Co-vertices: (6; 3) and (6; 1)
Find the right characteristics to the following ellipse.
Center: (-3; 3)
Foci: (-5; -3) and (11; -3)
Vertices: (-7; 3) and (13; 3)
Co-vertices: (-3; -3) and (-3; 9)
Write the following equation in standard form. Identify the related conic.
(1/16)(2/4)(3/2) = y - x
An ellipse
(1/16)(2/3)(2/2) = y - x
A circle
Write the following equation in standard form. Identify the related conic.
(1/4)(3/8)(1/2) = y - x
A hyperbola
(1/4)(3/8)(1/2) = y + x
An ellipse
Write the following equation in standard form. Identify the related conic.
A hyperbola
An ellipse
Write the following equation in standard form. Identify the related conic.
A hyperbola
A parabola
Write the following equation in standard form. Identify the related conic.
A hyperbola
A circle
Write the following equation in standard form. Identify the related conic.
A hyperbola
An ellipse
Write the following equation in standard form. Identify the related conic.
A hyperbola
An ellipse
Write the following equation in standard form. Identify the related conic.
A hyperbola
An ellipse
Write the standard form of the equation of a circle that passes through each set of points. Then identify the center and radius of the circle.
(x−5)²+(y−3)²=9, Radius = 3, Center (5; 3)
(x+3)²+(y−5)²=9, Radius = 3, Center (-3; 5)
Write the standard form of the equation of a circle that passes through each set of points. Then identify the center and radius of the circle.
(x−1)²+(y+7)²=16, Radius = 4, Center (1; -7)
(x+1)²+(y−7)²=4, Radius = 2, Center (-1; 7)
Write the standard form of the equation of a circle that passes through each set of points. Then identify the center and radius of the circle.
x²+(y−6)²=9, Radius = 3, Center (0; 6)
(x+6)²+y²=9, Radius = 3, Center (-6; 0)
Given x²=16y, state whether the parabola opens upward, downward, right or left, and state the coordinates of the focus.
The focus is located at (0; 4), The parabola opens upward
The parabola opens downward, The focus is located at (-4; 0)
Given (x−3)²=−8(y−2), state whether the parabola opens upward, downward, right or left, and state the coordinates of the vertex.
Vertex at (3; 2), The parabola opens downward
The parabola opens upward, Vertex at (2; 3)
Find the focus and directrix of the parabola whose equation is x²−6x+3y+18=0.
The focus is at (3; -15/4), The directrix is y = -9/4
The focus is at (15/4; -3), The directrix is x = 9/4
Find the focus and directrix of the parabola whose equation is (y−1)²=8(x+4).
The focus is at (-2; 1), The directrix is x = -6
The focus is at (-2; -5), The directrix is y = 6
Given the hyperbola 4x²−9y²=36, determine equations of the asymptotes, coordinates of foci.
y=2x/3 and y=−2x/3, Coordinates of foci at (−√13;0) and (√13;0)
y=4x/3 and y=−4x/3, Coordinates of foci at (−√5;0) and (√5;0)
Given the hyperbola 9x²−16y²=144, determine the right characteristics of hyperbola.
Endpoints of vertical axis: (0; -3) and (0; 3), Vertices: (-4; 0) and (4; 0)
Endpoints of horizontal axis: (5; -3) and (-3; -3), Foci at (0; 5) and (0; -5)
Given the hyperbola x²−16y²=64, determine the right characteristics of hyperbola.
Hyperbola opens horizontally, Vertices: (-8; 0) and (8; 0)
Hyperbola opens vertically, Endpoints of vertical axis: (0; -4) and (0; 4)
Given the hyperbola y²−4x²=4, determine the right characteristics of hyperbola.
Hyperbola opens vertically, Foci at (0; √5) and (0; -√5)
Hyperbola opens horizontally, Foci at (0; 5) and (0; -5)
Given the hyperbola 푦^2−4푥^2=4, determine the right characteristics of hyperbola.
Hyperbola opens vertically
Foci at (0; √5) and (0; -√5)
Vertices: (-2; 0) and (2; 0)
Endpoints of vertical axis: (0; -1) and (0; 1)
Select the equation of a line passing through the points (2,-4,1) and (0,4,-10).
2x - 4y + 11z = 0
(2, 4, 8) + t(2, 2, 11)
(2, 4) + t(2, 2)
4x + 10y - 2z = 0
Select the equation of a line that passes through the point (-7,2,4) and parallel to the line given by x=5-8t, y= 6+t, z=-12t.
74x - 2y - 8z = 0
(-7, 2, 4) + t(-8, 1, -12)
(-7, 1, 2) + t(-8, -1, -12)
724x - 4212y - 8z = 0
Determine the equation of a line that passes through the point P(1,2,-3) and parallel to the vector v = (4,-5,1).
(1, 2, -3) + t(4, -5, 1)
1 + 4t, 2 - 5t, -3 + t
(4, -5, 2) + t(1, 3, -1)
4x - 5y + z = 0
Determine the equation of a line that passes through the points P(0,7) and Q(5,0).
(0, 7) + t(5, -7)
5x - 7y = 0
(5, 0) + t(5, 7)
5x + 7y = 0
Determine the equation of a line that passes through the origin and is parallel to the vector v = (-2,3).
(0, 0) + t(-2, 3)
x = -2t, y = 3t
(1, 1) + t(-2, 3)
(2, 3) + t(-2, 3)
Find the slope of the line which equation is 2x−3y+9=0.
k = -2
k = 2
k = 2/3
k = -1/3
k = 3
Find the slope of the line, passing through the points A(1; 3) and B(2;5).
k = -2
k = 1/2
k = 2
k = -1/2
k = 0
Find equation of a line passing through the point A(1,3) and parallel to the line 4x+5y−5=0.
y = -4/5x + 11/5
5x−4y+7=0
4x+5y−19=0
y = 5/4x + 7/4
x+3y−5=0
Find equation of a line passing through the point A(1,3) and perpendicular to the line 4x+5y−5=0.
y = -4/5x + 11/5
5x−4y−7=0
4x+5y−19=0
y = 5/4x + 7/4
x+3y−5=0
Write canonical equation of a straight line passing through the point M=(1,0,2) and parallel to the vector l=(1,3,−1)
Write canonical equation of a straight line passing through the point M=(1,0,2) and parallel to the vector l=(0,3,−1)
Write canonical equation of a straight line passing through the point M=(0,0,2) and parallel to the vector l=(2,3,−1)
Find equation of a line which passing through the point (2; 0; −3) and parallel to the vector s=(2,−3,5)
Find equation of a line passing through the point (1; 0; 3) and parallel to the line {2x−y+3z−11=0 5x+4y−z+4=0
Find normal vector of the plane 2x+3y−5z+1=0
(2,−3,5)
(3,2,5)
(2,3,−5)
(2,−5,1)
(3,−5,1)
Find normal vector of the plane 3(x−2)+2(y−1)+4(z+1)=0
(2,1,−1)
(−6,−2,4)
(3,2,4)
(1,−1,1)
(3,1,−1)
Find normal vector of the plane −3y−5z+7=0
(−3,−5,0)
(0,2,5)
(0,−3,−5)
(−3,−5,7)
(3,−5,1)
Write equation of a plane, passing through the point (1,2,3) and perpendicular to the vector n=(0,2,3)
x+2y+3z−14=0
x+2y+3z−13=0
2y+3z−13=0
2y+3z−9=0
x+3z−9=0
Write equation of a plane, passing through the point (−1,6,0) and perpendicular to the vector n=(2,0,3)
2y+3z−14=0
−x+6y+2=0
2x+3z+2=0
2y+z−9=0
x+3z−9=0
Write equation of a plane passing through the point (0,2,−1) and perpendicular to the vector n=(4,2,1)
2y+2z−14=0
−x+6y+2z−9=0
4x+2y+z−3=0
2y−z−3=0
x+3z−9=0
Find equation of a plane, passing through the points (1; 1; 1) and (2;3;4) and perpendicular to the plane 2x−7y+5z+3=0
5y+7z−12=0
x+5y+7z−31=0
−31x−y+11z+21=0
2x−7y+5z=0
x+y+z−3=0
Find equation of a plane passing through the points (1; 1; 1),(0; −1;2) and (2;3;4)
5y+7z−12=0
5y+z−31=0
−2x+y+1=0
2x−y+5=0
−8z+4y+12=0
Find equation of a plane passing through the points (−1; 1; 1),(0; −1;2) and (2;−3;4)
y+7z−12=0
5y+z−31=0
x−z+2=0
2x−y−z+5=0
y+z−2=0
Find equation of a plane passing through the points (−1; 0; 1),(0; −1;2) and (5;−3;4)
y+z−12=0
5y+z−3=0
y+z−1=0
2x−z+5=0
y+z−2=0
Find equation of a plane passing through the points (−1; 0; 3),(0; −1;2) and (5;0;4)
y+z−12=0
x−y+z−3=0
x+7y−6z+19=0
2x−z+5=0
y+z−2x=0
Find equation of a plane passing through the points (0; 0; 3),(0; −1;2) and (5;0;4)
y+5z−12=0
x−5y+z−19=0
x+5y−5z+15=0
2x−z+5=0
5y+z−2x=0
Find the angle between two lines y=−2x−3 and y=x/2+1
0
π/4
π/2
π/3
π/6
Find the angle between two lines 5x−y+7=0 and 2x−3y+1=0
0
π/2
π/4
π/3
π/6
Find the angle between two lines 2x+y=0 and y=3x−4
0
π/2
π/4
π/3
π/6
Find the angle between two lines 3x+2y=0 and 6x+4y+9=0
0
π/2
π/4
π/3
π/6
Find the angle between two lines 3x−4y=6 and 8x+6y=11
0
π/2
π/4
π/3
π/6
Find the angle between two lines x^2−y^3=1 and x^3+y^2=1
0
π/2
π/4
π/3
π/6
Find angle between the line {x−2y+3=0 3y+z−1=0 and the plane −2x−y+3z+1=0
0
arccos(arccos(1/14))
π/2
arcsin(arcsin(1/14))
π/6
Find angle between the line {2x−2y+3=0 3y−z−1=0 and the plane −x−y+3z+1=0
0
arccos(arccos(3/√11))
arcsin(arcsin(7/11))
arccos(arccos(7/11))
π/6
Find angle between the line {x−2y+3=0 3y+z−1=0 and the plane −2x−y+3z+1=0
0
arccos(arccos(1/14))
π/2
arcsin(arcsin(1/14))
π/6
Find distance between the point P(1; 2) and the line 2x−y+3=0
1
0
3√5
1/2
-3/5
Find distance between the point P(1; 2; −1) and the plane 2x−y+z+3=0
1
0
2√6
1√6
3√5
Find distance between two parallel lines: 2x−4y+6=0 and x−2y+1=0
1
√5
2√5
1√2
3√5
Find distance between two parallel lines: x+y+3=0 and −2x−2y+9=0
1
3√2
15/2√2
1/√2
5√2
Find distance between two parallel lines: 푥+푦+3=0 and −2푥−2푦+9=0
1
3
2
Find distance between two parallel planes: 2푥−4푦−2푧+6=0 and푥−2푦−푧+1=0
2
5
4
6
1
Find distance between two parallel planes: 푥+푦−푧+3=0 and −2푥−2푦+2푧+9=0
15
2
1
3
2
Determine whether the line and plane are perpendicular or parallel or intersect at a point: {푥=−2−4푡 푦=3−2푡 푧=1+2푡 and 2푥+푦−푧=5
parallel
perpendicular
intersect at point
none of them
Determine whether the line and plane are perpendicular or parallel or intersect at a point: {푥=−5−4푡 푦=1−푡 푧=3+2푡 and푥+2푦+3푧−9=0
parallel
perpendicular
intersect at point
none of them
Determine whether the line and plane are perpendicular or parallel or intersect at a point: {푥=3푡 푦=1+2푡 푧=2−푡 and 4푥−푦+2푧−1=0
parallel
perpendicular
intersect at point
none of them
Determine whether the line and plane are perpendicular or parallel or intersect at a point: {푥=0 푦=푡 푧=푡 and 5푥−3푦+3푧−1=0
parallel
perpendicular
intersect at point
none of them
Determine whether the line and plane are perpendicular or parallel or intersect at a point: {푥=1 푦=푡 푧=푡 and 6푥+2푦−푧−3=0
parallel
perpendicular
intersect at point
none of them
Determine the equation of the plane that contains the pointsP(1,-2,0), Q(3,1,4) and R(0,-1,2)
2xyz−8+5=0
2(1)+8(2)+11=0
(1)(2)(4)=0
2xyz−8−5=0
Determine the equation of the plane that contains the points P(4,-3,1), Q(-3,-1,1) and R(4,-2,8)
14x+49y+7z=98
14x+49y+7z=0
14(5)+7(3)+7(1)=0
(1)(1)(2)=95
Determine the equation of the plane that contains the point (3,0,-4) and orthogonal to the line given by r(t) = (12-t, 1+8t,4+6t)
(3)−8(0)+6(4)=0
8x+5y+20z=0
7825xyz−=0
(3)(6)+6(4)=0
Find the equation of a plane containing the point (-8,3,7) and parallel to the plane given by 4x+8y-2z=45
4(8)+8(3)+2(7)=0
2x+6y+2z=21
2x+6y+4z=23
8(4)+3(8)+7(2)=0
Find the intersection between the line given by r(t) = (-2t,2+7t,-1-4t) and the plane given by 4x+9y-2z=-8
dy=8arctg(4x)/(1+16x^2)dx
(8, 10, 7)
dy=2arctg(4x)/(1+x^2)dx
dy=4arctg(4x)/(1+16x^2)dx
Find the intersection between the line r(t) = (4+t,-1+8t,3+2t) and plane given by 2x-y+3z=15
dy=6arcsin(3x)/(√(1−9x^2))dx
Lines do not intersect
(2, 1, 3)
dy=4arctg(4x)/(1+16x^2)dx
