Try to visualize the set of all points (x, y, z)in a coordinate space that are equidistant from the points P(0, 0, 0) and Q(0, 3, 0) . Use the Distance Formula to find an equation for this surface, and observe that it is a plane.

Q: (a) Find and identify the traces of the quadric surface x2 + y2 - z2 = 36 given the plane.

A:

Q: The surface described by the conditions 2 + y? + z² = 4, x 0 is the same as the surface described…

A:

Q: (b) Sketch the trace of the surface z = 2x? – y? when x = -1. Label your axes, and give the…

A: Given surface is  z=2x2-y2          (i) By putting x=-1 in (i), the surface becomes z=2-y2 The graph…

Q: Find parametrizations for the lines in which the planes x + y + z = 1, x + y = 2

A:

Q: Find the distance between the point and the plane. 19. (0,0,0) and 2x+3y+z12 20. (2,8,4) and 2x+…

A: a) given point is (0,0,0)  plane is 2x+3y+z=12 find the distance between the point and the plane  b)…

Q: Find the coordinates of the point where the line F = (1+ 2t)i + tj – 2tk intersects the plane 2x – y…

A:

Q: Consider the surface given by the equation z – y = x². Draw a total of at least 3 traces in at least…

A: It can be solved in 2nd part using simple geometry.

Q: The set of all points equidistant from the point (0, – 3, 0) and the plane y = 3 are represented by…

A:

Q: state the type of the quadric surface and describe the trace obtained by intersecting with the given…

A: Given: x2+y42+z2=1  ……1   Equation of plane is: z=14  ……2

Q: Let C be the curve of intersection of the parabolic cylinder x² = 2y, and the surface 3z = xy. Find…

A:

Q: If f(x, y, z) = V3x + y² – 2z, then find the slope of the surface f in the r- direction at the point…

A: Final answer is 3/2

Q: Consider the surface z = (x+ 1)² + (y – 2)² – 1 - Select a correct name for this surface and point…

A:

Q: Find an equation for the plane that is tangent to the surface z = In (x³ + y²) at the point…

A: Given:   A surface   z = ln x3 + y2   To Find :  An equation  for the plane that is tangent to the…

Q: Find the points on the surface represented by x2 + y? + z2 - 4x – 6y + 2z = -10 which are closest to…

A: To find - Find the points on the surface represented by x2 + y2 + z2 - 4x - 6y + 2z = -10 which are…

Q: Sketch and identify the surface given by the equation Z=: x² +

A:

Q: b) what is the direction 8. Find the equation of the plane tangent to the surface z? = x?y + y2x+ 10…

A:

Q: Match the surfaces (a)-(e) with the contour diagrams (1)-(V, (M)

A:

Q: Sketch the following planes in the window [0, 5] × [0, 5] × [0, 5]. The plane parallel to the…

A: First obtain the plane that parallel to the xz-plane containing the point (1, 2, 3).…

Q: Find the shortest distance between the surface z = x² + y² and the plane r + y – z = 4.

A:

Q: When the plane x = 3 passes through the surface x2 + y? + z² = 64, %3D the curve that forms from…

A:

Q: Find the surface area of the plane ( 15x- 5z+20y=35) if it is in the first octant

A: Solution:-

Q: b. z= 4x2 + 4y²

A: The equation describes surface of elliptic paroboloid.

Q: Show that the curve a(t) = (t,4,²) lies in a plane and find the equation of this plane.

A: Solution:- αt=t,1+tt,1-t2t t,1t+1,1t-t=x,y,z It is clear that, t-1t-1+1t-t=x-y+z x-y+z=-1

Q: Identify a generating curve on the XZ-plane for the surface of revolution with equation x? + y2 + z?…

A: Given: Surface of revolution with the equation x2+y2+z2=e2x To find: Identify a generating curve on…

Q: Find an equation for the surface consisting of all points P for which the distance from P to the…

A:

Q: Let (P) be the surface given by the following equation z = 5-x²-y² and (Q) be the surface given by…

A: Given surface P is, z=5-x2-y2    ----(1) For xy projection, put z=0 in (1) we get, 0=5-x2-y2⇒…

Q: Find an equation for the surface consisting of all points that are equidistant from the point (0,-…

A: Point (0, -3,0) Y= 3

Q: Find an equation of the surface consisting of all points P(r, y, z) that are twice as far from the…

A: To find an equation of the surface whose all points are twice as far from the plane z=-4 as from the…

Q: the trace of the surface in the plane x=1?

A:

Q: 2? = a? + 2y?

A: Consider the given surface z2=x2+2y2 The trace in the plane x=k are z2-2y2=k2 For k=0, z2-2y2=0⇒…

Q: Find an equation for the surface obtained by rotating the line z = 2x about the z-axis. + x²) b. z?…

A:

Q: Sketch and label the level surfaces of g(r, y, z) =1-r² – y² for g = 0,-1,-2.

A:

Q: For the surface 4x² + y² −z = 0, classify the indicated trace as an ellipse, hyperbola, or parabola.…

A: We have been given the equation  4x² + y² −z = 0           (1)

Q: Show that the minimum distance from the origin to a point on the plane ax + by + cz = d is |d| Ja? +…

A: Given :               ax + by + cz = d         We know that minimum distance (p) of a point x1,y1,z1…

Q: Find the equation for the sphere centered M(2,-3,2) tangent to the D:6x-3y+2z-11=0 plane

A: we have to find the equation for the sphere centered 2,-3,2 and tangent to the plane 6x-3y+2z-11=0.

Q: Find the extreme values of the function ƒ(x, y, z) = xy + z2 on the circle in which the plane y - x…

A:

Q: Let C be the intersection curve of the parabolic cylinder √2x2 = 2y and the surface 3z =√2xy. Find…

A:

Q: Write an equation for the plane tangent to the surface F(x, y, z) = 0 at the point (a, b, c).

A: Given surface is Fx, y, z=0 and the point is a, b, c.

Q: Show that the curve a(t) = (t, , ) lies in a plane and find the equation of this plane.

A: For all 't' , we have α'(t) = 1 , -1t2 ,-1 - 1t2α"(t) = 0 , 2t3 , 2t3 Now  α'(t) × α"(t) = 2t3 ,…

Q: Try to visualize the set of all points (x, y, z)in a coordinate space that are twice as far from the…

A: a) Suppose the distance between two points  A(x, y, z)  and the origin   Plug into the distance…

Q: Locate and classify the stationary points on the surface z=xy2-2xy-2x2-3x, remembering that  x, y ,…

A: Solving for stationary points of given function using partial derivatives

Q: ). Find a parametric representation for the line of intersection of the following two planes: I - y…

A: We use cross product to find the direct vector for the line.

Q: Find parametrizations for the line in which the planes 3r- 6y – 2z = 3, and 2x+y-2z = 2 intersect.

A: Given Equation of planes 3x-6y-2z=3 and 2x+y-2z=2

Q: Show that the curve al lies in plane and b) finel the equation of the plane that contains the curve…

A:

Q: Find the equations of planes that just touch the sphere (x−5)^2+(y−3)^2+(z−1)^2=9 and are parallel…

A: Solution  Given (x−5)^2+(y−3)^2+(z−1)^2=9 (a)  Parallel to the  xy-plane:   ( z-1)2= 9   z - 1= | 3…

Q: The surface that is linear with the three variables, x, y, and z is called cylinder plane sphere O…

A:

Q: Find an equation of the plane that is normal to the surface Iz + yz? = 2+ yz at the point (2,-1, 1).

A:

Q: x² + 9y² – 2² = 9

A: The given equation is: x2+9y2-z2=9           (1)

Q: Find the distance between the origin and the intersection line of the planes : 2x+y+z=2 and x+2y+z=3…

A:

Q: Find the distance from the point B = (1,0, 2) to the plane 2P whose general equation isx + y - z =…

A: Given: equation of plane is x+y-z=1  We know that distanced from the pointx1,y1,z1 to the…

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.