Consider two straight lines described by the following parametric equations x = 3+t, y = 4 + 3t, z = 1+2t; x = -2 + 2s, y = 5 – 2s, z = 1– s respectively, where t and s are parameters. (a) Find the point of intersection of these two lines. (b) Find the acute angle between these two lines.

Q: Without eliminating the parameter t, find a Cartesian equation for the line tangent to the curve C,…

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Q: 33 and Complete the parametric equations for the line where the planes – 3x – 9y – 12z = - 11x + 2y…

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Q: Complete the parametric equations for the line where the planes - 5x – 2y + 2z = 2 and - 13x + 10y +…

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Q: Question 2. (a) Prove that the lines with parametric equations: x = 1+ t, y =1+6t, z = 2t and x = 1+…

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Q: Find the exact length of the curve described by the parametric equations. x = 5 + 6t², y = 3 + 4t³,…

A: Find the length of the curve   x= 5+t2 ,y= 3+4t2 , 0<=t<=1 .

Q: Find a set of parametric equations for the rectangular equation that satisfies the given condition.…

A: Given :                y = 2 x - 5  ,  t = 0        at point (3,1)

Q: let L1 and L2 be the lines whose parametric equations are x-3 z-1 L1 == y - 4 = 1 =yー4= -= y – 2 4 х…

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Q: Describe the relationship between the lines with the following parametric equations: x = 1- 41, y =…

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Q: Eliminate the parameter to write the parametric equations as a Cartesian equation. If x(t) = 3t – 2,…

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Q: The lines L1 and L2 are determined by the following symmetric and parametric equations,…

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Q: Find the distance between the point (1, −2, 4) and the line given by the set of parametric equations…

A: The given point is M=(1, −2, 4) The direction vector of the line is =s=(2,1,2) A point on the line…

Q: Sketch the plane curve represented by the parametric equations x = Vi and y = 2t – 1 by eliminating…

A: Sketch the plane curve represented by the parametric equations             x=t     and     y=2t-1 by…

Q: A curve is defined by the parametric equations z (t) = e-* and y (t) = e*. What is in terms of t ? A…

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Q: Determine the direction angles for the line with parametric equations x = 5 + 2t, y = 12 - 8t, and z…

A: Direction angle

Q: The parametric equation of the line through the origin that is parallel to the line given by x = t,…

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Q: Consider two non-parallel lines , l1 & l2, in R³ with parametric equations l1: P+ tu l2: Q + tv.…

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Q: 2. Find the point(s) of intersection of the following two parametric curves, by first eliminating…

A: From x=t+5 we get t=x-5 Then we plug this in the y=t2 So, the equation is: y=(x-5)2   We can plug…

Q: Find the (perpendicular) distance from the line given by the parametric equations æ(t) 3 – 4t %3D…

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Q: Find the inflection points of the curve defined by the parametric equations:x=t2,y=t+t3.

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Q: Find the point at which the line given by the parametric equations below intersects the plane.…

A: Given a system of linear equations are, 2X + 4Y -3Z =-48.................(1) X = 10 +…

Q: 18. What is the rectangular equation of the curve whose parametric equations are x = 2t, y= 2+ 3t*?…

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Q: Sketch and describe the orientation of the curve given by the parametric equations. x = V7, y = 2t –…

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Q: The angle and the parametric equations for the intersect line between planes a + y = 1, and 2r+y-2z…

A: Given, x+y=1, 2x+y-2z=2 we know angle between two plane is…

Q: Complete the parametric equations for the line where the planes - 7x + y + 3z = 3 and 4x7y + z = 23…

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Q: Find the angle between the two planes x- 2y + z = 29, 2x + 3y – 2z = 6. And also find the parametric…

A: here we will find angle between two planes and parametric equation of plane's line of intersection,

Q: Without eliminating the parameter t, find a Cartesian equation for the line tangent to the curve C,…

A: Let's find.

Q: Find the parametric equation for the line where the planes 4x – by + 8z = - 36 and – by + z = – 15…

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Q: Find the rectangular equation for the following two parametric equations. x = 2t – 1 and y = t² + 6…

A: Given equation is  x = 2t-1            ..............(i) And y = t2+6          .............(ii) So,…

Q: At which point will the plane 2x + 4y – 3z = -48 and the line given by the parametric equations x =…

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Q: Complete the parametric equations for the line where the planes - 7x + y + 3x = 3 and 4x7y + z = 23…

A: Given two planes -7x+y+3z=3 and 4x-7y+z=23

Q: What is the length of the curve defined by the parametric equations æ (t) = t² – 2t and y (t) = t³ –…

A: If x=f(t) and y=g(t)are parametric equations of the curve where t ranges from t1 to t2then…

Q: Find the parametric equation of the intersection of the planes x + y = 1 and x + z = 2.

A: Equation of plane 1 - x+y = 1        Normal vector of plane 1 = n1 = 1,1,0…

Q: Consider the parametric equations x = √t − 2 and y = (1/ 2)t + 1, t ≥ 2. What is implied about the…

A: Consider the given parametric equation as x=t-2 and y=12t+1

Q: x = 1 + 4t,    y = 3 − t2

A: Here we extract t from the 1st equation then put this into second equation.

Q: Find the distance between the point (2, 3, 2) and the line given by the set of parametric equations…

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Q: Find the parametric equation for the line where the planes 7x – 6y + z = 10 and 2x + y – 3z = 4…

A: We have, r(t) = (0, -2, -2) + t(17, 23, 19) The detailed solution is as follows below:

Q: Find the length of the parametric curve x = 2t + 1, y = 3 + 2tž, 0 < t <%.

A: Find the Length of the curve

Q: Let L1 and L2 be the lines whose parametric equations are L1 :x= -63 +9t,y = 10 – 21, z = 45 – 6t L2…

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Q: Find the parametric equation for the line where the planes 4x – 6y + 8z = - 36 and – 6y + z = – 15…

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Q: Complete the parametric equations for the line where the planes -4x + 2y – 14z = – 32 and -x – y –…

A: Use parametric equation of tangent line

Q: Sketch the curve represented by the parametric equations x = e−t ,y = e2t − 1 (indicate the…

A: The given parametric equations of curves are: x = e^(-t) y = e^(2t) - 1 The graphs of these…

Q: Which of the following is the parametric equation of the line through the point (1,2,3) and parallel…

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Q: Allow A be the point with Cartesian coordinates (6, 9), and D be the plane curve with parametric…

A: The slope of the tangent to the curve: y=fx at x=a is f'a. The point-slope form of a line that…

Q: Sketch the plane curve represented by the parametric equations x = Vi and y = }t + 1 by eliminating…

A: The parametric equations are given by x=t   …1 and y=12t+1   …2 Take the square of the equation (1)…

Q: Complete the parametric equations for the line wherekche planes 2=0and-2z + 2y + 2z -4 intersect.…

A: The equation of the planes are: P1: 2z = 0 ⇒ z = 0                                  .........1 P2:…

Q: Find a parametric equation for the line that is perpendicular to the graph of the equation 2x + y…

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Q: Find the (perpendicular) distance from the line given by the parametric equations ¤(t) -2 – 2t y(t)…

A: Solution: The objective is to find the perpendicular distance

Q: Find two different sets of parametric equations for the rectangular equation  y = 4x + 1, t = −1 at…

A: Given equation is y=4x+1, t=-1 and the point is (-2,-7).

Q: Find the distance between the point (4, −1, 5) and the line given by the set of parametric equations…

A: The point is: P(4, −1, 5) The line is: x = 3, y = 1 + 3t, z = 1 + t The line is: &lt;3, 1, 1&gt; +…

Q: Convert the line determined by the symmetric equations x – 5 y +1 z 3 to parametric form. a(t) y(t)…

A: Given symmetric equation of the line is x-55=y+13=z-9 which we need to convert into parametric form

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.