The paraboloid z= 6-x-x2 -2y2 intersects the plane x=1 in a parabola,Find parametric equations for the tangent line to this parabola at the point (1,2,-4) . Use a computer to graph the paraboloid,the parabola,and the tangent line on the same screen. tangent line on the same screen.

Q: Use the Distance Formula to show that the equation of the parabola with focus F(0, 1) and directrix…

A: Given that Focus of parabola is F0, 1. Directrix is y=-1. We have to prove that y=14x2 is the…

Q: Find parametric equation for the lines .. The line through (0, -7, 0) perpendicular to the plane x +…

A: A parametric equation of a line can be written if a point on the line is given and a vector parallel…

Q: In each part, find an equation of the sphere with center (5, -6, –3) and satisfying the given…

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Q: Determine the parametric equations of the curve of intersection of the ellipsoid 4x² + y² + 42² = 4…

A: To find: Parametric equation of the Intersections of the curves  4x2+y2+4z2=4 and y2=x2+2z2.

Q: Find a set of parametric equations for the parabola whose equation is y = 9 - x2.

A: Given equation is, y=9-x2

Q: 3. Determine the orthogonal trajectory of the family of parabolas x² – y + 32x = 0. %3D Sketch some…

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Q: Use the Distance Formula to show that the equation of the parabola with focus F(0, 2) and directrix…

A: Given:            Focus of parabola is : F(0,2)Directrix is:​y=−2.

Q: Find parametric equations for the line through Po = (3, –1, 1) perpendicular to the plane 8x + 5y –…

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Q: Find the parametric equations for the line in which the planes -4x +y+3= = 5 and 2x+y+ = 3…

A: Given that:-4x+y+3z=5    .............(1)2x+y+z=3          ..............(2)

Q: Find a parametric form for the line through the point (8, –6, –3) and perpendicular to the plane 2x…

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Q: state whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an…

A: Rearrange the equation, 3x2-7y2=z by dividing both sides by 21 and obtain x27-y23=z21. The general…

Q: Find parametric equations for the complete parabola x = y2. Answersare not unique.

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Q: Find an equation of the line in parametric form that passes through the vertex of the paraboloid…

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Q: Find parametric equations and a parameter interval for the motion of a particle starting at the…

A: Given,

Q: Find a parametric equations representation for the curve (x + 3) 2 . (y – 2)² 1 4

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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: In each part, find an equation of the sphere with center (1, – 7, –8) and satisfying the given…

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Q: Find parametric equations for the line through (-2, 0, 4) parallel to v = 2i + 4j - 2k

A: we know that parametric equation of line which passes through point x0,y0,z0 and parallel to vector…

Q: State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an…

A: 3x2-7y2=z On simplification, we get ⇒z=x213-y217⇒z=x2132-y2172⇒z=x132-y172

Q: Find the point P of intersection between the line x(t) = (-3,-2, 1) + t (1, 1, -3) and the plane 4z…

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Q: 3. Find a set of symmetric and parametric equations of the line passing through the point (1, 4…

A: This is the problem from the vector algebra and 3D geometry.

Q: On a separate piece of paper, sketch the graph of the parabola y = x + 3. On the same graph, plot…

A: Given: Equation of parabola, y=x2+3

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 parametric equations and a parameter interval for the motion of a particle that starts at (a,…

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Q: Find a parametric equation for the line that is perpendicular to the graph of the equation 3x + 4y +…

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Q: Find parametric equations of the line containing the .point (2, 3,-1) and normal to the plane x+…

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Q: Find a parametric equation for the line that is perpendicular to the graph of the equation 2x + 4y +…

A: Equation of line normal to graph of functions f(x) is :  r<t> = <x0,y0,z0> +…

Q: Find parametric equations, in terms of t, of the line passing through (2, 2, 12) and parallel to the…

A: Given: The line is passing through 2,2,12. Also, it is parallel to the line: x=5-9ty=2z=4+t The…

Q: Try to sketch by hand the curve of intersection of the circu- lar cylinder x + y² = 4 and the…

A: Given, Circular cylinder, x2+y2=4 Parabolic cylinder, z=x2

Q: Find the parametric equations of the line segment through the origin and perpendicular to the plane…

A: Given: Given equation of the plane is x-2y+3z=5 We want to find the parametric equations of the line…

Q: (i) Find the acute angle (in degrees) between the following two planes:  8x+6y+12z=24 and…

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Q: Find the parametric equation of the line that passes through the point (4, -5, 0) and is…

A: We have to find the parametric  equation of the line.

Q: Complete the parametric equations of the line through the point (4,-8,3) and perpendicular to the…

A: See this solution in below

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: find a parametric equation for the line that is perpendicular to the graph of the given equation at…

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Q: In each part, find an equation of the sphere with center (5, –8, –4) and satisfying the given…

A:

Q: Find parametric equations for the line in which the planes x + 2y + z = 1 and x - y + 2z =-8…

A: x + 2y + z = 1 and x - y + 2z =-8 are the two planes.Let z=t So, the equations become:

Q: Find the parametric equations for the line through the point (5, 2,9) and perpendicular to the plan…

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Q: Find the parametric equations for the line of intersection of the planes x+y-z = -20 and 9.x + 3y +…

A: Given planes, x+y-z=-20 and 9x+3y+z=16. The objective is to find the parametric equations for the…

Q: Find an equation of the tangent line to the hyperbola defined by (5x^2)+(4xy)−(2y^2)−4x=195 at the…

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Q: Find a parametric equation for the line that is perpendicular to the graph of the equation 4x + y +…

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Q: Find parametric equations for the line tangent to the curve of intersection of the surfaces at the…

A: Given, the surfaces:               x+y2+2z=4,   x=1  Point: (1, 1, 1) Let         z = 1-2t

Q: Sketch the curve represented by the parametric equations (indicate the orientation of the curve). x…

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Q: y+3x-6z=8

A: answer is in next step

Q: Find a set of parametric equations for the curve of intersection of a paraboloid x² + (y − 2)² = z…

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Q: Prove that using the only the form of the given equation (no graphing) that it is a paraboloid.…

A: We need to prove that the the equation: x2+z2-y4=1 is a paraboloid.

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

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Q: Find parametric equations for the line through (8,4,6) parallel to the x-axis. X = y = - o <t< ∞

A: The equation passing through point (8,4,6). The parametric equation passing through point (8,4,6)…

Q: The angle and the parametric equations for the intersect line between planes x + y = 1, and 2x + y–…

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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.