Show that the tangent plane to the quadric surface at the point (x0, y0, z0) can be written in the given form. Ellipsoid: x2/a2 + y2/b2 + z2/c2 = 1Tangent plane: x0 x/a2 + y0 y/b2 + z0z/c2 = 1

Q: The area of the plane z = 2x + 1 above the disk of radius 1 centered at the origin is (а) 4л. (b)…

A: As we know, z=fx,y defines a surface in xyz-space.  Formula for the surface area of the surface…

Q: parametrizes the portion of the sphere x² + y² + z² = 9 that is above the xy-plane and below the…

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Q: Find the point on the surface z=x2-y2 at which the tangent plane is parallel to the plane…

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Q: Find the average height of the hemispherical surface z = sqrt(a2 - x2 - y2 )above the disk x2 + y2…

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

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Q: Evaluate || f (x,y,z) dS where S is the portion of the sphere centered at the origin of radius two…

A: we have the equation of sphere x2+y2+z2=22s is the portion bounded between xy plane and z=1 in the…

Q: Consider x=h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent…

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Q: Find the tangent plane to the surface z = cos(3ry) + z' + 2y at the point (1,0). (z = cos(3ry)+z'+…

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Q: Consider points P and Q in the x-y plane, with P = (1,0) and Q = (0,1). The line integral %3D Q…

A: The line joining P1,0 and Q0,1 is     x-11-0=y-00-1=t(say)⇒x=t+1,y=-t⇒dx=dt,dy=-dt

Q: If (a,B.y) is a point at which the surface x2+y2-z²-2x+12=0 has a horizontal tangent plane, then y|…

A: As per guidelines, we will solve first question only. Let α,β,γ be the point on the surface…

Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

Q: Sketch the surface z = 3x? + y? +1 and find its linear approximation at P-(0, -1)

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Q: Find the centroid of the portion of the sphere x2 + y2 + z2 = a2 that lies in the first octant.

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Q: Consider x = h(y, z) as a parametrized surface in the natural way. Write the equation of the tangent…

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Q: Suppose f(x, y) satisfies the basic existence and uniqueness theorem in some rectangular region Rof…

A: Consider the given: Let the function f(x,y) satisfies the basic existence and uniqueness theorem in…

Q: Let S be the passage of R ^ 2 defined by the parabola with equation y = 18-x ^ 2 and by the line…

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Q: (b) Give an example of a parametrisation of the unit circle, C = {(x, y) E R² | x² + y² = 1}, that…

A: The parameterization of a circle  that generates the clockwise orientation of C is given by: x=r…

Q: PICTURE

A: We are given that the cone ax2+by2+z2 = 0 and the ellipsoid 2x2+4y2+z2 = 42 intersect and are…

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

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Q: Find the area of the surface of the part of the plane 3x + 2y + z = 6 that lies in the first octant.

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Q: Let G be the solid in the first octant bounded by the sphere 2x? + 2y? + 2z? = 32 and the coordinate…

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Q: Suppose that C is the curve of interesection between the plane z = x and the cylinder x² + y? be the…

A: Consider the given function. z=x and x2+y2=1 As, the function S be the interior of this ellipse…

Q: Let C be the positively oriented circle centered at the point zo with radius r>0. Use a…

A: We will first parametize the circle C

Q: Find the equation of the tangent plane for the surface z = xy at the point (-1,2,-2) O z=x-2y+2…

A: The equation of tangent plane for the surface z =f(x,y ) at point (m,n,f(m,n)) is given as:

Q: Let E be the solid ellipsoid bounded by the surface y? = 1, a2 62 c2

A: Given that, the solid ellipsoid bounded by the surface x2a2+y2b2+z2c2=1 The volume of the unit…

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 the curvature of a superellipse defined by the parametric equations…

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Q: . Show that the tangent plane to the surface 2yyo cz at the point P(xo,Y0, zo) is given by the 2rro…

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Q: Let C be an ellipse in a horizonal plane lying above the xy-plane. Which type of quadric surface is…

A: Given, Let C be an ellipse in a horizontal plane lying above the xy-plane.

Q: Find all points on the surface xy - z2 = 1 which are closest to that origin. Let x,y,z be the…

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Q: Find a parametrization c(θ) of the unit circle such that c(0) = (−1, 0).

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Q: Compute the surface area of the ellipse that lies on the plane z = 11 – 3x + 2y and inside the…

A: The surface area of the surface given by z=f(x,y) where (x,y) is a point from the region D in the…

Q: If a parametric surface given by r1(u, v) = f(u, v)i + g(u, v)j+ h(u,v)k and –1< u<1, –2 < v< 2, has…

A: We will find out the required value.

Q: Consider the surface z = 1? – y?. Let Tp denote the tangent plane at a point P on this surface. Find…

A: Given equation of surface in the form of z=f(x,y) so first write in the more general form of…

Q: Find an equation of the tangent plane to the surface given implicitly by z= sin(zy) at the point (,…

A: A tangent plane to a surface is a plane that just touches the surface at a single point. The…

Q: Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface…

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Q: Find an equation of the tangent plane to the given parametric surface at the specified point. x = u…

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Q: Find the average height of the hemispherical surface z = √(a2 - x2 - y2) above the disk x2 + y2 ≤ a2…

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Q: Find symmetric equations for the tangent line to the curve of intersection of the paraboloid zx +y…

A: Given  The equation of Paraboloid Z=x2+y2 The equation of Ellipsoid 2x2+y2+z2=76

Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(1,1,3). Which of the following…

A: tangent planes to a surface are planes that just touch the surface at the point and are “parallel”…

Q: If the tangent plane to the surface z = +y is parallel to the plane r+ 2y-z 0 at the point (1, a,…

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Q: Find the tangent plane to the elliptic paraboloid z = x² – 2y at the point (1,1,1). - O A. z = 2x –…

A: Given that: z=x2-2y2      and point is:   (1, 1, 1)

Q: Find the distance between two planes a and a1 that are tangent to the ellipsoid 2 +2y? + z² = 5/8…

A: We have to find the distance between the two tangent planes parallel to x+y+z=0  that are tangent to…

Q: b) Use Stokes' Theorem to evaluate fF•dr for F(x,y,z) =-yi+ xz j+ zk where C is the curve of…

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Q: Find a set of parametric equalions for the surface formed when the curve y(1+z²) – 1 = 0 is revolved…

A: Find a set of parametric equations for the surface formed if the curve y1+z2-1=0 is revolved about…

Q: A tangent plane to the ellipsoid x². y? a b2 meets the axes in A, B, C. Show that the envelope of…

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Q: Show that the tangent plane to the surface x 2 a 2 + y 2 b 2 = cz at the point P(x0, y0, z0) is…

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Q: Find the plane tangent to the surface z=x+ y at the point (3,4,25). a. 6x+8y-z=25 b. 6x-8y-z=25 c.…

A: We have to find the plane tangent for the given surface

Q: The surface area of the part of the paraboloid z=x+y that lies under the plane z=2 is: Z=,

A: Surface area is given as A=∬D1+δzδx2+δzδy2dA

Q: Find any points on the hyperboloid  x2 − y2 − z2 = 6  where the tangent plane is parallel to the…

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