Consider the planes below: ⊼(pie): x+2y–1z=0 ⊼: –x+2y–3z–4=0 ⊼: x+6y–5z–4=0 a) Without solving the system, and by examining the normals, determine the nature of the intersection of the three planes. Include diagrams with answers.  b) Find the intersection of the planes

Q: Use Cartesian coordinates to evaluate JIJ y² dv 2 D where D is the tetrahedron in the first octant…

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Q: Calculate , z² dS if S is the prism whose base is the triangle in the limited xy plane by the x axis…

A: Here given figure is Since prism has 5 surfaces (1)-OAB (2) OADE (3) CDE (4) OBCD (5) ABCE

Q: The distance between the centers-of the following two spheres: x²-22x+y2-58y+z?=-961,…

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Q: Find the area of the band cut from the paraboloid x2 + y2 - z = 0 by the planes z = 2 and z = 6.

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Q: Consider the planes below: π1: 2x+3y−z+2=0 π2: x−5y+6z+1=0 π3: 3x−2y+5z+3=0 a) Without solving…

A: According to the problem, we have

Q: Find whether the planes below intersect, and if so, where they intersect using Gaussian elimination.…

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Q: a) Show that the distance between the parallel planes Ax + By + Cz = D1 and Ax + By + Cz = D2 is |D1…

A: (a) To show that the distance between two parallel planes Ax+By+Cz=D1 and Ax+By+Cz=D2 is…

Q: Find the dimensions of the largest rectangular box in the first octant (x, y, z > 0) with one corner…

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Q: Is v = 2i - 4j + k related in any special way to the plane 2x + y = 5? Give reasons for your answer.

A: Given:

Q: Evaluate (x2 + y? + z)dS, where S is the portion of the plane z = x + 2 that lies inside the…

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Q: Consider the planes below: ⊼(pie)1: x+2y–1z=0 ⊼2: –x+2y–3z–4=0 ⊼3: x+6y–5z–4=0 a) Without…

A: To solve the following problem

Q: a) Evaluate (-ri-yj+2z k) n dS, where a is the portion of the paraboloid : 3x +3y between the planes…

A: We have to Evaluate the Integral using polar coordinate

Q: Consider the planes below: ⊼1(pie): x+2y–1z=0 ⊼2: –x+2y–3z–4=0 ⊼3: x+6y–5z–4=0 a) Without…

A:

Q: a) Evaluate (-ri-yj+2z k) -n dS, where o is the portion of the paraboloid z = 3r + 3y? between the…

A:

Q: Let W be the region that is cut out from r² + y² < 4 by the paraboloid z = r² + y² and the plane z =…

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Q: a) Find the points at which the twisted cubic F(t) = ti + t} +tk intersects the plane 4.x +2y+z = 24…

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Q: Find the angle (in degrees) between the planes 4y + z = 4 and – 4x + 2z= - 3.

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Q: Find the distance between the planes M: 3x – 4y + z – 17 = 0 and N: 6x – 8y + 2z + 68 = 0.

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Q: Find the point p(x, y, z) on the plane 2x + y - z - 5 = 0 that is closest to the origin.

A: Given:The given plane is 2x+y-z-5=0 that is closest to the origin.To find:we have to find…

Q: 2. Do the three planes x + 2y + z = 4, y - z = 1, x+ 3y= 0 have at least one commom point of…

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Q: The base of a solid is the region in the xy-plane between the the lines y = x, y = 4x, x = 1 and x =…

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Q: Describe the intersection of the horizontal plane z = h and thehyperboloid −x 2− 4y 2 + 4z 2 = 1.…

A: Put z=h in given equation -x2-4y2+4h2=1⇒x2+4y2-4h2=-1

Q: a) Evaluate [-ri-yj+2: k) - n ds, (-a i- where a is the portion of the paraboloid z = 3r + 3y…

A:

Q: 5. Use Cartesian coordinates to evaluate ƒƒƒ v³² av where D is the tetrahedron in the first octant…

A: Given problem:-

Q: Find the surface area of the portion of the part of the paraboloid z = 3 – x² – y² that lies above…

A: Given z = 3-x2-y2 , above the plane z=-1

Q: 9-The distance d between the parallel planes -2x+y+z 0, 6x-3y-3z-5-0 is: a) V54 b) 54 V5 d) 3 c) e).…

A: Distance between two parallel lines

Q: Use Cartesian coordinates to evaluate JIJ y² dV D where D is the tetrahedron in the first octant…

A:

Q: Given: planes II₁ : 2x - 5y + z = - −1 and II₂ : x + 5y + z = 5 intersecting on a line l (c) Give a…

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Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: Find for the point P(x,y,z) on the plane 3x + y – z – 15 = 0 that is closest to the origin.

A: The distance of the point P(x,y,z) from the origin is: d2=x2+y2+z2 Now we have to minimize d if the…

Q: Find the average height of the paraboloid z = x2 + y2 over the square 0&lt;= x &lt;= 2, 0…

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Q: 2. Do the three planes X + 2y + z = 4, y - z = 1, x+ 3y = 0 have at least one commom point of…

A: The equation of the plane is given by the formula ax+by+cz+d=0. The system of planes can be solved…

Q: Find the area of the cap cut from the paraboloid y2 + z2 = 3x by the plane x = 1.

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Q: Find the surface which intersects the surfaces of the system z(x+ y) =c(3z+1) orthogonally and…

A: To finding the surface which is intersecting given system and orthogonal to circle is given as…

Q: // 2° zdS, where S is the part of the plane z = 4+ 2æ + y above the rectangle [0, 5] × [0, 3]…

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Q: Evaluate € fff.₂²². e² dV where E is enclosed by the paraboloid z = 3 + z² + y², the cylinder z² +…

A: We need to find integral.

Q: Find the distance between the following parallel planes: (a) 3x – 4y +z=1 and 6x – 8y + 2z =3 (b) –…

A: Solve the above questions.

Q: The surface area of the cap cut from the paraboloid y + z? = 3x by the plane x = 2 is Select one or…

A: Given the paraboloid  y2+z2=3x  we have to find the surface area of the cap cut from the paraboloid…

Q: Among all rectangular solids defined by the inequalities 0≤x≤ a, 0sysb, 0szs1, find the one for…

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Q: Given two surfaces x²y + z = 3 and xlnz – y² = -4, a) calculate the angle between the normals to…

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Q: a) Show that the line I = 0, y = t, z =t where te R a) lies in the plane 6z+4y–4x = 0 b) is parallel…

A: In the question it is asked to solve for the planes having equation given below.

Q: Find the distance between the planes M: 3x – 4y + z – 17 = 0 and N: 6x – 8y + 2z + 68 = 0. %3D

A: Distance between parallel planes d=(d2-d1) /(sqrt(a2+b2+c2)

Q: Find the distance from the point of intersection of liies ( X=t, y=3-3t, =-224) and (x=1+t y=4rt,…

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Q: a) Evaluate /| (-ri-yj+2: k) · n dS, where o is the portion of the paraboloid z = 3x2 + 3y? between…

A: Define a function fx,y=z or fx,y=3x2+3y2.  So, the flux integral is:…

Q: The distance from a point S(xo, Yo, Zo) to a plane M: ax + by + cz + d = 0 is given by In PS| d :…

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Q: Consider the planes below: x+2y+3z+3= 0 2x-y+z+1=0 TI3: 5x+2y+7z+7=0 a) Without solving the system,…

A: Given:π1: x+2y+3z+3=0π2: 2x-y+z+1=0π3: 5x+2y+7z+7=0 To Find: a. Without solving the system, and by…

Q: e|| x°zdS, where S is the part of the plane z = 1+ 4x + 2y above the rectangle [0, 4] x [0, 2]…

A:

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: a) Evaluate (-x i - yj+ 2z k) - n dS, where o is the portion of the paraboloid z = 3r2 + 3y? between…

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Q: a) Evaluate //(-ri-yj+2: k) n dS, where a is the portion of the paraboloid : = 3r - 3y between the…

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.