Introduction to Parametric Equations

Key Questions

• How do you find the parametric equations of a curve?

  It depends on the curve you're analyzing,

  In general, finding the parametric equations that describe a curve is not trivial. For the cases that the curve is a familiar shape (such as piecewise linear curve or a conic section) it's not that complicated to find such equations, due to our knowledge of their geometry.

  As an example of a curve that's difficult to find the parametrization, just close your eyes and draw any curve on a piece of paper. Chances are that curve has a complicated description, and even if you found the equations describing it, it would probably be hard to work with them.
  For problems that have these kind of difficulties, it's interesting to use numerical methods (such as piecewise linear approximations).

  An example that's not comprised of combinations of line segments or conic sections:

  The cycloid is a curve with many interesting properties. Finding the parametric equations that describe it is very useful.

  A cycloid is the curve generated by rotating a circle (of, say, radius #a#) over the #x# axis while tracing the path of the point initially at the origin.
  The parameter adopted will be the angle described by the point intially at the origin, the center of the circle and the point of contact between the circle and the #x# axis.

  

  Basic geometry* will give us the equations:

  #x(theta)=a[theta-sin(theta)]#
  #y(theta)=a[1-cos(theta)]#

  *The #x# coordinate is given by the difference between the arclenght #a theta#, wich is the distance between the origin and the projection over the #x# axis of the center of the circle, and the projection over the #x# axis of the radius of the circle associated with the tracing point, equal to #a sin(theta)#. The #y# coordinate is given by the difference between the height of the center of the circle, equal to the radius #a#, and the projection over the #y# axis of the radius of the circle associated with the tracing point, equal to #a cos(theta)#.

  Vinicius M. G. Silveira · 1 · Dec 16 2014
• How do you find the parametric equations for a line segment?

  The line segments between #(x_0,y_0)# and #(x_1,y_1)# can be expressed as:
  #x(t)=(1-t)x_0+tx_1#
  #y(t)=(1-t)y_0+ty_1#,
  where #0 leq t leq 1#.

  The direction vector from #(x_0,y_0)# to #(x_1,y_1)# is
  #vec{v}=(x_1,y_1)-(x_0,y_0)=(x_1-x_0,y_1-y_0)#.
  We can find any point #(x,y)# on the line segment by adding a scalar multiple of #vec{v}# to the point #(x_0,y_0)#. So, we have
  #(x,y)=(x_0,y_0)+t(x_1-x_0,y_1-y_0)#,
  which simplifies to:
  #(x,y)=((1-t)x_0+tx_1,(1-t)y_0+ty_1)#,
  where #0 leq t leq 1#.

  Wataru · · Sep 6 2014
• How do you sketch the curve with parametric equations #x = sin(t)#, #y=sin^2(t)# ?

  Since we know that #-1 le sint le 1#, the curve is limited to #-1 le x le 1#.

  By plugging #x=sint# into #y=sin^2t#, we have

  #y=(sint)^2=x^2#.

  Hence, the curve is the portion of the parabola #y=x^2# between #x=-1# and #x=1#, which looks like this:

  

  I hope that this was helpful.

  Wataru · 1 · Oct 5 2014
• What are parametric equations used for?

  Parametric equations are useful when a position of an object is described in terms of time #t#. Let us look at a couple of example.

  Example 1 (2-D)

  If a particle moves along a circular path of radius r centered at #(x_0,y_0)#, then its position at time #t# can be described by parametric equations like:

  #{(x(t)=x_0+rcost),(y(t)=y_0+rsint):}#

  Example 2 (3-D)

  If a particle rises along a spiral path of radius r centered along the #z#-axis, then its position at time #t# can be described by parametric equations like:

  #{(x(t)=rcost),(y(t)=rsint),(z(t)=t):}#

  Parametric equations are useful in these examples since they allow us to describe each coordinate of the position of a particle separately in terms of time.

  I hope that this was helpful.

  Wataru · · Oct 5 2014

• How do you find the parametric equation of a parabola?
• How do you find the parametric equations for a line segment?
• How do you find the parametric equations for a line through a point?
• How do you find the parametric equations for the rectangular equation #x^2+y^2-25=0# ?
• How do you find the parametric equations of a circle?
• How do you find the parametric equations of a curve?
• What are parametric equations used for?
• What is the parametric equation of an ellipse?
• How do you sketch the curve with parametric equations #x = sin(t)#, #y=sin^2(t)# ?
• How do you find the vector parametrization of the line of intersection of two planes #2x - y - z = 5# and #x - y + 3z = 2#?
• How do you use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces #x^2+y^2=25# and #z=3x^2#?
• How do you write #x = e^t-1# and #y=e^2t# as a cartesian equation and then sketch the curve?
• How do you graph parametric equations?
• For #f(t)= (t^2,t^3)# what is the distance between #f(1)# and #f(3)#?
• For #f(t)= (t^2,t^3)# what is the distance between #f(1)# and #f(3)#?
• For #f(t)= (t-t^3,t^3)# what is the distance between #f(0)# and #f(3)#?
• For #f(t)= ( t-2, sqrtt -t )# what is the distance between #f(0)# and #f(1)#?
• For #f(t)= (t/(t+3), t/sqrt(t+1) -t )# what is the distance between #f(0)# and #f(1)#?
• For #f(t)= (t/(t-3), t/sqrt(t^2+1) )# what is the distance between #f(0)# and #f(1)#?
• For #f(t)= (1/(2t-3), t/sqrt(t^2+1) )# what is the distance between #f(0)# and #f(1)#?
• For #f(t)= (1/(2t-3), te^t )# what is the distance between #f(0)# and #f(1)#?
• For #f(t)= (e^(t^3-t^2), te^t )# what is the distance between #f(0)# and #f(1)#?
• For #f(t)= (lnt/e^t, e^t/t )# what is the distance between #f(1)# and #f(2)#?
• For #f(t)= (lnt/e^t, te^t)# what is the distance between #f(1)# and #f(2)#?
• For #f(t)= (lnt-e^t, t^2e^t)# what is the distance between #f(1)# and #f(2)#?
• For #f(t)= (lnt-e^t, t^2/e^t)# what is the distance between #f(2)# and #f(4)#?
• For #f(t)= (lnt/t, t^2/e^t)# what is the distance between #f(2)# and #f(4)#?
• For #f(t)= ((lnt)^2/t,t^3)# what is the distance between #f(2)# and #f(4)#?
• For #f(t)= (te^(3t),t^2-t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (te^(1-3t),2t^2-t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (t/e^(1-3t),2t^2-t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (e^t/t,4t+1/t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (1/t,-1/t^2)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (1/(1-t),-1/(1+t^2))# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (sint-cost,t^2)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (sint-cost,t)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (sin^2t,t/pi-2)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (sin^2t,cos^2t)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (sint,cost)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (sint,cos^2t/t)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (sin2t,cos^2t)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (cos2t,sin^2t)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (cost/2,sin^2t)# what is the distance between #f(pi/4)# and #f(pi)#?
• For #f(t)= (sqrt(t+2)/(t+1),t^2+3t)# what is the distance between #f(0)# and #f(2)#?
• For #f(t)= (sqrt(t)/(t+1),t^2-t)# what is the distance between #f(0)# and #f(2)#?
• For #f(t)= (t/sqrt(t+1),t^2-t)# what is the distance between #f(0)# and #f(2)#?
• For #f(t)= (te^(t-1),t^2-t+1)# what is the distance between #f(0)# and #f(2)#?
• For #f(t)= (1/(t-3),t^2)# what is the distance between #f(0)# and #f(2)#?
• For #f(t)= (t^3-t^2+1,t^2-t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (t^3-t+1,t^2-t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (t/(t+2)-t+1,t^2-t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (t+21,-3t^2-2t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= (t-2,-t^2-2t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= ((t-1)^2,-t^2-2t)# what is the distance between #f(2)# and #f(5)#?
• For #f(t)= ((t+3)^2,t^3-t)# what is the distance between #f(2)# and #f(4)#?
• How do you find the parametric equations for the line through the point P = (3, 5, -2) that is perpendicular to the plane −5x + 1y + 3z = 1?
• How do write parametric equations for Margot's position t seconds after s if Margot is walking in a straight line from a point 30 feet due east of a statue from a point 24 feet due north of the statue and walks at a constant speed of four feet per second?
• How do you graph the curve whose parametric equations are given and show its orientation given #x = sqrt{t} + 4#, #y = sqrt{t} - 4#, where #t>=0#?
• How do you find the set of parametric equations for the line in 3D described by the general equations x-y-z=-4 and x+y-5z=-12?
• How do you find a vector equation for the line through the point (5, 1, 4) and perpendicular to the plane x - 2y + z = 1?
• How to find the point where the line x = –1 – t, y = 2 + t, z = 1+ t intersects the plane 3x + y + 3z = 1 ?
• Question #9eda6
• How do you find the equations of the tangents to that curve #x=3t^2 + 1# and #y=2t^3 + 1# that pass through point (4,3)?
• Question #6fe41
• Consider the point #P(3,−5, 1)# and the line #L ∶ x = 2t − 1 , y = −t + 2 , z = −2t; −∞ < t < ∞#. a. Find the equation of the plane passing through P perpendicular to L. b. Find the equation of the plane passing through P and containing L?
• Consider the line which passes through the point P(-1, -5, 4), and which is parallel to the line x=1+5t y=2+5t z=3+5t. How do you find the point of intersection of this new line with each of the coordinate planes?
• How do you find parametric equations for the line through (2, 4, 6) that is perpendicular to the plane x − y + 3z = 7?
• How do you find vector parametric equation for the line through the point P=(−4,−5,3) perpendicular to the plane 3x−4y+3z=−1?
• How do you get a initial and terminal point given #x=cos(pi-t)#, #y=sin (pi-t)#, #0<=t<=1#?
• How do you graph parametric equations x(t)=3t and y(t)=2-2t on [0,1]?
• How do you find the range given #x=t^2-4# and #y=t/2# for #-2 <= t <= 3#?
• How do you find the range given x=3-2t and y=2+3t for -2 ≤ t ≤ 3?
• Consider the parametric equation: #x = 15(cos(theta) + (theta)sin(theta))# and #y = 15(sin(theta) - (theta)cos(theta))#, What is the length of the curve for #theta = 0# to #theta = pi/8#?
• A curve is given by the parametric equations: #x=cos(t) , y=sin(2t)#, how do you find the cartesian equation?
• How do you find a vector equation and parametric equations for the line segment that joins P to Q where P(-7, 2, 0), Q(3, -1, 2)?
• What is the parametric equation of the line through (1,1,1) that is parallel to i+j+k?
• How do you find a vector equation and parametric equations in t for the line through the point and perpendicular to the given plane. (P0 corresponds to t = 0.) P0 = (5, 0, 8) x + 2y + z = 9?
• How do you find parametric equations and symmetric equations for the line through t0 and parallel to the given line t0 = (4, -2, 4) and x + 1 = y/2 = z + 5?
• How do you find the parametric equations for the line through the point P = (2, -2, -1) that is perpendicular to the plane 1x + 3y - 2z = 1?
• How do you determine scalar, vector, and parametric equations for the plane that passes through the points A(1,-2,0), B(1,-2,2), and C(0,3,2)?
• How do you find the set of parametric equations for the line in 3D described by the general equations x-y-z=-4 and x+y-5z=-12?
• How do you find the set of parametric equations for the line through the point P(2,-1,0) which is perpendicular to the plane described by the general equation -x+z=2?
• Given #y=(x/5)^2# how do you derive a parametric equation?
• Given #x^2 + (y – 2)^2 = 4 # how do you derive a parametric equation?
• What is parametric equation of the line created by the intersecting planes x = 2 and z = 2?
• A curve in the xy-plane is defined by the parametric equations #x = t^3 + 2# and #y = t^2 - 5t# how do you find the slope of the line tangent to the curve at the point where x = 10?
• How do you find parametric equation for given curve: Line of slope 8 through (-4,9)?
• How do you find parametric equation for given curve #y²=4ax#?
• How do you find the parametric equation for the line which pass by two points p1=(1,-2,1) p2=(0,-2,3)?
• How do you determine the vector and parametric equations for the plane through point (1, -2,3) and parallel to the xy-plane?
• Do the parametric equations #x = 3t^3 + 7#, #y = 2 - t^3#, #z = 5t^3 + 1# determine a line?
• How do you convert the parametric equations into a Cartesian equation by eliminating the parameter r: #x=(r^2)+r#, #y=(r^2)-r#?
• How do you convert #x^2 + y^2 = 49# into parametric equations?
• How do you find parametric equations for the line which passes through the point (1,−2,3) and is parallel to both of the planes 3x + y + 5z = 4 and z = 1 − 2x?
• The surface #z=xsqrt(x+y)# intersects the plane y=3 along a curve c, how do you find the parametric equations for the tangent line to this curve at the point P(1,3,2)?
• How do you find the parametric equations for the line through the point P = (4, -4, 1) that is perpendicular to the plane 3x + 1y - 4z = 1?
• How do you find the parametric equations of the line of intersection of the planes x+2z=0 and 2x-3y+4?
• How do you find parametric equation for the line through the point P(-7,-3,-6) and perpendicular to the plane -2x + 2y + 6z =3?
• How do you determine the parametric equations of the path of a particle that travels the circle: #(x−3)^2+(y−1)^2=9# on a time interval of #0<=t<=2#?
• The curve given by the parametric equations #x=16 - t^2#, #y= t^3 - 1 t# is symmetric about the x-axis. At which x value is the tangent to this curve horizontal?
• How do you write the parametric equations represent the ellipse given by #x^2/9 + y^2/81=1#?
• How do you write an equation in slope-intercept form of the line with parametric equations: x=2+3t and y=4+t?
• How do you find a vector parametric equation r(t) for the line through points P=(-3,-1,1) and Q = (-8,-4,5) If r(6) = P and r(10) = Q?
• How do you find parametric equations for the tangent line to the curve with the given parametric equations at the point #( 1, 4, 4)#. #x = cos(t), y = 4e^8t, z = 4e^ -8t#?
• How do you write a vector equation and a parametric equation for each line: the line through A(1,-3,1) and parallel to vector u=(2,-2,1)?
• How do you write a vector equation and a parametric equation for each line: the line through A(3,0,4) and parallel to the x-axis?
• How do you write a vector equation and a parametric equation for each line: the line through A(-1,2,1) and B(1,2,1)?
• How do you find the parametric and symmetric equation of the line passing through the point (2,3,4) and perpendicular to the plane 5x+6y-7z=20?
• How do you find the angle between the planes 2x+5y-z=6 and 3x-2y+6z=10?
• How do you give parametric equations for the line through (1, 3, -5) that is perpendicular to the plane #2x -3y +4z = 11#?
• How do you find the parametric equations for the tangent line to the curve #x=t^4−1#, #y=t^2+1#, #z=t^3# at the point (15, 5, 8)?
• How do you find parametric equations for the line of intersection of the planes 2x + 5z + 3 = 0 and x -3y + z + 2 = 0?
• How do you find the parametric equations of the line that passes through #A=(7,-2,4)# and that is parallel to the line of intersection of the planes #4x-3y-z-1=0# and #2x+4y+z-5=0#?
• A curve C is defined by the parametric equations: #x=t^2# and #y = t^3-3t#, how do you show that C has two tangents at the point (3,0) and find their equations?
• How do you find the equations of tangent lines at the point where the curve crosses itself #x=t^2-t# and #y=t^3-3t-1#?
• How do you find two different sets of parametric equations for the given rectangular equation y = 1/x?
• How do you find the parametric equation through the point (6, 0, 10) and parallel to the plane -3x+2y-4z = 0?
• Eliminating the parameter in the parametric equations x = 5 - 3t, y = 2+t yields what equation?
• Given the parent function #f(x) = x^2#, how do you write a set of parametric equations to represent the function?
• Let #x = e^t + 3# and #y = e^(2t)+6e^t+9# how do you eliminate the parameter and write y in terms of x?
• How do you write the cartesian equation for x = t - 2 and y = -(t²) + t + 1?
• Let l be the line that passes through the point P=(−5,−7,−1) and is perpendicular to the plane 8x−6y−1z=15, how do you find the parametric equations?
• How do you convert #(y = 2 + x)# to parametric equation?
• How do you find parametric equations for the tangent line to the curve with the given parametric equations at the specified point #x = 1+10 * sqrt(t)#, #y = t^5 - t#, and #z=t^5 + t# ; (11 , 0 , 2)?
• How do you find parametric equations for the line of intersection of two planes 2x - 2y + z = 1, and 2x + y - 3z = 3?
• How do you find the equation of the tangent line to the curve given by parametric equations: #x=1+(1/t^2)#, # y=1-(3/t)# at the point when t=2?
• How do you write the corresponding rectangular equation by eliminating the parameter given x=3t-1, y=2t+1?
• How do you write the corresponding rectangular equation by eliminating the parameter #x = t^2 + t# and #y = t^2 - t#?
• How do you find parametric equations and symmetric equations for the line through the points (1, 3, 2) and ( -4, 3, 0)?
• How do you find the (shortest) distance from the point P(1, 1, 5) to the line whose parametric equations are x = 1 + t, y = 3 - t, and z = 2t?
• Consider the parametric equation #x = 9(cost+tsint)# and #y = 9(sint-tcost)#, What is the length of the curve for #t= 0# to #t=3pi/10#?
• How do you find the parametric equations for the intersection of the planes 2x+y-z=3 and x+2y+z=3?
• How do you convert each parametric equation to rectangular form: x = t - 3, y = 2t + 4?
• How do you convert each parametric equation to rectangular form: #x = t^(3/2) + 1#, #y = sqrt{t}#?
• How do you combine the parametric equations into one equation relating y to x given x=4cost and y=9sint?
• How do you find the equation of the tangent line for the curve given by x = 2t and #y = t^2 + 5# at the point where t = 1?
• How do you find parametric equations for the line through the point (0,1,2) that is perpendicular to the line x =1 + t , y = 1 – t , z = 2t and intersects this line?
• How do you find a parametric equation for a particle moving twice counter-clockwise around the circle #(x-2)^2 + (y+1)^2 = 9# starting at (-1,-1)?
• How do you find two difference parametric equations for the rectangular equation y=8x-7?
• Consider the parametric equations x = 3t - 5 and y = 2t + 3 how do you eliminate the parameter to find a Cartesian equation of the curve?
• How do you find the parametric equations for the rectangular equation #x^2+y^2=16#?
• How do you find the parametric equations for the line through the point (2,3,4) and perpendicular to the plane 3x + 2y -Z = 6?
• How do you find parametric equations and symmetric equations for the line through #(3, −2, 5)# and parallel to the line #x + 3 = y/2 = z − 2#?
• How do you show that the curve x=cos t , y= sin t cos t has two tangents at (0,0) and find their equations?
• How do you find parametric equations to represent the line segment from (-3,2) to (1,-8)?
• How do you change the parametric equation to cartesian equation #x = t^2 - 1# and #y = t^3#?
• How do you find parametric equations for the line through P-naught=(3,-1,1) perpendicular to the plane 3x+5y-7z=29?
• Given Parametric equations: #x = 2(t)^2# and y = 4t how do you find the cartesian equation?
• How do you find parametric equations for the path of a particle that moves along the #x^2 + (y-3)^2 = 16# once around clockwise, starting at (4,3) 0 ≤ t ≤ 2pi ?
• How do you find parametric equations for the path of a particle that moves along the #x^2 + (y-3)^2 = 16# Three times around counterclockwise, starting at (4,3) 0 ≤ t ≤ 6pi?
• How do you find parametric equations for the path of a particle that moves along the #x^2 + (y-3)^2 = 16# Halfway around counterclockwise, starting at (0,7). 0 ≤ t ≤ pi?
• How do you write parametric equations for the curve that polar equation #r = 3cos(theta)#?
• A line is defined by the parametric equations: x = cos2t and #y = sin^2t# how do you find the cartesian equation of the line?
• How do you find the vector equation and parametric equations for the line through# (0, 14, -10)# and parallel to the line:#x = -1+2t, y = 6-3t, z = 3+9t#?
• Given x = 2 cot t and #y = 2 sin^2 t# for #0<t<= pi/2# how do you find the cartesian equation for this curve and state the domain?
• How do you rewrite the two parametric equations’ as one Cartesian equation x = sec(t), y = tan(t), #-pi/2<t<pi/2#?
• How do you find the Cartesian equation from given parametric equations #x = (2t )/ (1+t^4)# and #y = (2t^3) / (1 + t^4)#?
• How do you find parametric equations for the path of a particle that moves around the given circle #x^2 + (y – 2)^2 = 4# clockwise, starting at (2, 2)?
• The motion of the particle is given parametrically by #x(t)=3t^2-3#, #y(t)=2t^2# for t is greater than or equal to 0, how do you find the speed of the particle at t=1?
• Given x(t)=3sin(t) - 3, y(t)=t-1 for 0 is less than or equal to t is less than or equal to 2pi How do you find the position of the particle at t=3?
• Given x(t)=3sin(t) - 3, y(t)=t-1 for 0 is less than or equal to t is less than or equal to 2pi How do you find the velocity of the particle at t=3?
• Given x(t)=3sin(t) - 3, y(t)=t-1 for 0 is less than or equal to t is less than or equal to 2pi How do you find the speed of the particle at t=3?
• What is #r sin theta = 5# in #(x,y)# coordinates ?
• Question #cad9c
• How do you convert parametric equation to cartesian x = t - 2 and y = -(t²) + t + 1?
• What does the graph #r = sqrt(sintheta)# look like in plane polar coordinates? How do you graph it?
• How do you find the Cartesian equation of the curve with parametric equations #x=2cos(3t)#and #y=2sin(3t)#, and determine the domain and range of the corresponding relation?
• Question #71d19
• What is the area bounded by the parametric equations? : # x=acos theta # and # y=bsin theta #
• How would you solve something like #asint=bt# or #acost=bt# for #t#?
• Question #df769
• Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter?
• 19. "Describe the motion of a particle with position (#x,y#) as #t# varies in the given interval" ?
• Solve for #dy/dx#?