1. Let P be the plane defined by z = a for some constant a. (a) For which two numbers a = a1, a2 does P intersect the unit sphere in a single point? (b) For which number a is the radius of the circle where P intersects the unit sphere the largest? What is this largest radius?

Q: For numbers 1-2, consider a plane bounded by the curves y=9-x² and y=x+3. 1. What is the desired…

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Q: Which of the following planes is tangent to the sphere x² + y + z' = 17 at the point (2, -2, 3)? %3D…

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Q: If a parametric surface given by r₁ (u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -2 ≤ u ≤ 2, -3 ≤ v ≤…

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Q: (4) Find the points Po = (xo, Yo, 2o) on the sphere x? + y? + z² = 9 so that the tangent plane to…

A: Topic- equation of tangent in 3D

Q: What is the tangent plane equation of sphere f(x, y. :)=x² + y² += =6 at point (1,-1,2)? O a.…

A: I hope that this solution will help you .Thankyou. Best regards.

Q: 6 Consider points P and Q in the x-y plane, with P = (1,0) and Q (0,1). The line integral %3D %3D Q…

A: Introduction :Given, two points P and Q .P= (1,0)Q= (0,1)We have asked for the line integral of

Q: (A) Determine the plane that contain the point A(2,5,1) and the line L, t=x-2, 3t =y-1, t=z+1

A: The equation of the line L given in the parametric form, that is, Thus, the above equation of the…

Q: 4. Consider the curve in the plane defined by r3+y3 = 6xy. %3D (a) Find for any point (r, y) on the…

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Q: Let P be the plane that is tangent to the equation xyz + 260 = z' + 30x + 26y at the point (7,7,4)…

A: The tangent plane to the surface is a plane that only touch the surface at the point.

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: 3. Consider a plane going through the point p -1 and with normal n = 4 Write down an algebraic…

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Q: 3. Find the plane determined by the intersecting lines L: x = -1+t, y = 2+t, z = 1- t; L2: x = 1 -…

A: Since you have posted multiple questions in this question which are not interlinked so I have solved…

Q: The curve intersects the plane (in a three dimensional space) y+z = 2 at the point ( (Hint: First,…

A: Topic- intersection point of curve with Plane

Q: (3) Consider the quadric surface SC R, having equation a2 + y? – 22 = 9. Find all lines through P =…

A: The quadratic equation is given by x2+y2-z2=9.  General equation of line is: x,y,z=a,b,c+tl,m,n…

Q: 1. Find the area of the surface the part of the plane 3x + 2y + z = 6 that lies on the first octant

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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: 6. In Section 12.5, you learned that the distance from a point (ro, Yo, zo) to a plane ax+by+cz = d…

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Q: Draw a picture of two spheres of different sizes being tangent to each other at exactly one point P…

A: For part 1 we are given two spheres of different sizes i.e different radius which are tangent to…

Q: Let R = (c, 1,-1) where c denotes a real number. Let ä denote the plane x + y – z = 2. Find the…

A: Given Information -  The distance from the point R = (c,1,-1) to the plane x+y-z = 2 is 2. We have…

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: (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: Find the projection in the ry plane of the curve of intersection of the surfaces 222 = r² + y? and r…

A: Find your answer below

Q: 16. Let F(r, y, z) = ryi+azj+yzk and C be the curve of intersection between the sphere a? + y² + 2²?…

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Q: (a) Two surfaces are called orthogonal at a point of intersection if their normal lines are…

A: To show that two vectors are orthogonal, show that the dot product is zero.

Q: AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of…

A: Given in a circle of radius r, there are two chords AB and AC such that AB=2AC. Also, the distance…

Q: Suppose the solution set of x1 + x2+2x4 What are the equations of the plane at (1,2,3,4) which is…

A: First we find direction vectors of the plane of solutions.

Q: Find the extreme values of ƒ(x, y, z) = x2yz + 1 on the intersection of the plane z = 1 with the…

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Q: 12. a. Use vectors to show that the distance from P;(x1, Y1, Z1) to the plane Ax + By + Cz = D is…

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Q: Find all planes which (i) are tangent to the elliptic paraboloid z = x² + y², and (ii) pass through…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

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: Show that the surfaces x2 + 2y2 + 3z2 = 3,   x2 + y2 + z2 + 6x − 10y + 14 = 0, are tangent to each…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: (a) Two surfaces are called orthogonal at a point of intersection if their normal lines are…

A: a Given surfaces Fx,y,z=0 and Gx,y,z=0 Then their partial derivatives are Fx,Fy,Fz and Gx,Gy,Gz Let…

Q: Find the projection in the ry plane of the curve of intersection of the surfaces 2z2 = r? + y? and…

A: Let's find.

Q: [b] Find the distance from a point P(x₁, y₁, 2₁) to the plane Ax+By+Cz = D. Follow this hint:- Take…

A: Given: The equation of plane is Ax+By+Cz=D and the point is Px1, y1, z1. We have to find the…

Q: a) Evaluate Fự ds, where the curve C is the intersection between the surface z = V1- y? and the…

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Q: 1. Find a parametric representation for the surface: the part of the sphere x2 + y? + z2 = 4 that…

A: According to the given information, It is required to calculate the parametric representation for…

Q: If a parametric surface given by ri (u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -3 <u < 3, –4 < v<…

A: We solve this

Q: 56 Consider points P and Q in the x-y plane, with P = (1,0) and Q = (0,1). The line integral %3D Q…

A: Line integrals

Q: 3. Show that every sphere through the circle x2 +y-2ax+r²=0, z=0 cuts orthogonally every sphere…

A: This is a co-ordinate geometry problem

Q: Find the extreme values of ƒ(x, y, z) = x2yz + 1 on the intersection of the plane z = 1 with the…

A: consider the following equations  f(x,y,z)=x2yz+1                                  (1)h(x,y,z)=z-1…

Q: Which of the following represents a plane surface whose x-, y-, and z-intercepts are a, b, and c…

A: Given questions are solved.

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

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Q: Find the point lying on the intersection of the plane  x + (.5)y + (.25)z = 0  and the sphere  x2…

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

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Q: (3) Consider the quadric surface S C R, having equation a2 + y? - z2 = 9. Find all lines through P…

A: General equation of line from point 3,-3,3 is x,y,z=3,-3,3+tl,m,n.  Substitute it in the quadratic…

Q: 7. Find the area of the portion of the plane z = 2y that lies directly above the circle x +(y-1)² =1…

A: we have to find area of the portion of plane .and coverting the problem in polar coordinate makes it…