Find the point on the curve of intersection of the paraboloid x² • y² + 2z=4 and the plane x-y+ 2z -0 that is closest to the origin O A (1, 1, 1) O B. (1, -1, 1) O C(1,1, 1) O D. (1, 1, -1)

Q: Find the point closest to origin in R³ that is on both the planes 3x+y+z=5 and x+y+z=1.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: An equation of a plane P passes through the point (0,0,-3) and perpendicular to the line L:x = 1-2t,…

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Q: C is the hexagon in the xy-plane with vertices (5, −5), (5, 5), (0, 10), (−5, 5), (−5, −5), and (0,…

A: Given : Vertices : (5,-5) , (5,5), (0,10), (-5,5), (-5,-5) and (0,-10) Integral given :…

Q: Sketch the surfaces CYLINDERS z = y2 - 1

A: We need to sketch the surface formed by the function

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 extreme values of ƒ(x, y, z) = x( y + z) on the curve of intersection of the right circular…

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Q: Integrate G(x, y, z) = xyz over the triangular surface with vertices (1,0, 0), (0, 2, 0), and (0, 1,…

A: To Determine: integrate Gx,y,z=xyz over the triangular surface with vertices (1,0,0),(0,2,0) and…

Q: Find a parametrization of the paraboloid given by f (x, y) = x 2 + y2.

A: To find a parameterization of the paraboloid given by: fx,y=x2+y2

Q: Calculate dz – yz dy – sen(2)dz, where Cis the path connecting the point A(-2, 2, 4) to the point…

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Q: The points on the surface x2−2y2−3z2 = 33 at which the tangent plane is parallel to the 8x+4y+6z=5…

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Q: 3. Find the area of the part of the surface x 2z + y° - 3, that lies above the triangle in the…

A: Given, x=2z+y2-3, i.e. z=12x-y2+3. As it is known that the surface area is given as,…

Q: Show that the system of confocal and co-axial parabolas y²-4a(a+x) is self orthogonal, where a is…

A: If the two curves in the family cut each other at the right angle, then they are considered as self…

Q: Find the orthogonal trajectories of the family of curves: Ellipses with centers at (0,0) and two…

A: Find the orthogonal trajectories of the family of curves: Ellipses with centers at (0,0) and two…

Q: The point on the plane 2x+y+z=1 that is closest to the point (1, 2, 3) is (a, b, c). Then a+b-2c= O…

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Q: É = ( 2x³ + y³, y³ + 2³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 –…

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Q: C is the intersection curve between the paraboloid z=x²+y² and

A: Given that C is the intersection curve between the paraboloid z=x2+y2 and the cylinder x2+y2=1.

Q: Find an equation for the plane that is tangent to the surface z= In (x²+y^) at the point P(0,1,0).…

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Q: find the trace of the plane in the given coordinate plane.  3x + 4z = −2, xy

A: To find the trace of the plane in the given coordinate plane  3x+4z=-2 --- (1) in the xy plane

Q: (a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

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Q: The tangent plane to the elliptic paraboloid z = 2x2 + y² at the point (1, 1, 3) is %3D 1) 8x + 6y +…

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Q: (a) Show that the two surfaces S₁: z = xy and S₂ : z = ³x² - y² intersect perpendicularly at the…

A: As per our guideline, we are supposed to solve only first question. Kindly repost other questions as…

Q: Sketch the surfaces HYPERBOLOIDS (y2/4) - (x2/4) - z2 = 1

A: we have to sketch the surface of the hyperboloids  y24-x24-z2=1

Q: Find a parameterization for the hyperbolic paraboloid z = z – y².

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Q: Find an equation of the plane tangent to the surface z = x2 - y at the point (2, 1, 3). O A. 4x - 2y…

A: Find an equation of the plane tangent to the surface z= x2 - y2 at the point (2,1,3). Here i…

Q: The portion of the surface z = 2x + y2 that is above the triangular region with vertices (0, 0), (0,…

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Q: (B) Integrate paraboloid (x, y, z)=- 2x-²-2-0 by the plane x=1. over the surface cut from the end of…

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Q: Find /| F-dS if F(x, y, z) = 2xi+2yj+2zk and S is the part of the paraboloid z = = 8-x?-y? lying…

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Q: 28. Show that the ellipsoid 2x² + 3y² + z² = 9 and the sphere %3D x? + y? + z? - 6x – 8y – 8z + 24 =…

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Q: Find an equation of the plane tangent to the surface xy z2 = 2 at the point (2, 1, -1). О А.х+ бу +…

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Q: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.

A: we have to find the volume

Q: The tangent plane equation of the level surface z =-x² – 3y² at point (2,2,-16) is. O a.…

A: Given: The tangent plane equation of the level surface z=-x2-3y2 at point (2,2,16)

Q: Find the area of the surface x2 - 2y - 2z = 0 that lies above the triangle bounded by the lines x =…

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Q: Find the point on the surface r – 2xy + 2y² – x+ y = 0 closest to the point (1, 2, –3).

A: Consider the provided question, We have to find the point on the given surface closest to the point…

Q: Find an equation of the plane through the origin that is parallel to the plane 5x – 4y + 6z + 10 =…

A: Here we have to find the equation of a plane.

Q: É = ( 2x3 + y°, y3 + z³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 –…

A: We will find out the required flux.

Q: Use Lagrange multipliers to find the maximum and minimum of f(r, y, z) = -r + 2y + 2z on the ellipse…

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Q: (b) Integrate (x, y,z)=(y+z)+1 the paraboloid 4x - y-z -0 by the plane x 1. over the surface cut…

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Q: Find %3D as if F(x, y, z) = 2xi+2yj+2zk and S is the part of the paraboloid z = 4-x-y lying above…

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Q: Find the points on the hyperboloid 9x^2  - 45y^2 + 5z^2 = 45 where the tangent plane is parallel to…

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Q: The part of the surface z =1  + 3x+ 2y 2 that lies above the triangle with vertices (0, 0), (0, 1)…

A: Part of the surface: z=1+3x+2y2∂z∂x=3, ∂z∂y=4yVertices of the triangles are (0,0) , (0,1) , (2,1)

Q: Find the equations of the system of curves on the cylinder 2y = x² orthogonal to its intersections…

A: The given family of curves is got as the intersection of F(x,y,z)=2y-x2=0 with a one-parameter…

Q: Find an equation for the plane that is tangent to the surface z = In (x² + y) at the point P(0,1,0).…

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Q: Find a parametrization for the line segment beginning at P, and ending at P2. P (2,-3,2) and P2(0, –…

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Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

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Q: 11. Evaluate ([(y² +z? )dS where S is the part of the paraboloid x=4-y -? that lies in front of the…

A: We will evaluate the given integral.

Q: Find the absolute maximum and absolute minimum of the function f (x, y, z) = x³ + 2y along the curve…

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Q: Find the point lying on the intersection of the plane x + + z = 0 and the sphere x2 + y2 + z2 = 25…

A: According to the given information, it is required to find a point lying on the intersection of the…

Q: Sketch the surfaces HYPERBOLOIDS z2 - x2 - y2 = 1

A: Sketch the surfaces HYPERBOLOIDS z2 - x2 - y2 = 1

Q: Find a generating curve on the xz-plane for the surface of revolution with equation x²+ y²+ z²= e2x

A: Solution: Given equation is x^2+y^2+z^2=e^(2x)

Q: 12. Find the distance from the point P(0,0,0) to the plane given by x + y +z= 6.

A: