Let S be the hyperboloid x 2 + y 2 = z2 + 1 and let P = (a, {3, 0) be a point on S in the (x, y )-plane. Show that there are precisely two lines through P entirely contained in S (Figure 18). Hint: Consider the line r(t) = (a +at, f3 + bt, t) through P. Show that r(t) is contained in S if (a, b) is one of the two points on the unit circle obtained by rotating (a, {3) through ± ~. This proves that a hyperboloid of one sheet is a doubly ruled surface, which means that it can be swept out by moving a line in space in two different ways.

Q: Let Z = (1, 1, 2) be a point in R³. Find the closest point to Z tha lies on the plane give by -x+y-z…

A: The normal vector to the plane is <-1, 1, -1>. The point you seek would have to be some…

Q: Let L be the line segment with endpoints at (1,2,3) and (-1,1,4). A light source at is at the point…

A: We will find shadow of given points on the plane given and then find parametric equation of the line…

Q: Let C be the positively oriented closed plane curve consisting of the line segment from (-3,0) to…

A:

Q: 2. Let D be the half-disc in the xy-plane given by x +y <9. 2. Calculate e-?+v}dA Select one: a.…

A: Let  D  be the half-disc  in the xy planegiven  by  x2+y2&lt;9calculate ∫∫D  e-x2+y2  dA

Q: Show that the line [x, y, z]= [5,-1,3]+[4,-5,2] lies on the planes 3x+2y-z = 10 and 7x+4y-4z=19.

A:

Q: Find a plane through P (2, 1, 1) and perpendicular to the line of intersec- tion of the planes:…

A: find a plane through P2,1,1 and perpendicular to the line of intersection of the planes  2x+y-z=3 ,…

Q: Let l be the line given parametrically by x = (1, 3) + t(-2, 1), tЄR. Which of the following points…

A: A parametric equation is an equation that employs an independent variable  to define the dependent…

Q: Find the plane through P = (4, −1, 9) containing the line r(t) =〈1, 4, −3〉+ t〈2, 1, 1〉.

A: The plane contains the line r(t)=1, 4, -3+ t2, 1, 1, so a vector in the plane is 2, 1, 1. The line…

Q: e) Plane that is parallel to the xy-plane that contains the point A(2, 1, 3) (ca

A:

Q: Find the distance between the line 2x - 3y + z = 6, 2y- z = 4, and the plane passing through the…

A:

Q: Let line l : x=y/2=z/3 and point P(1,0,1). Find an equation of the plane (in point-normal form)…

A: Solution of given question is written in below steps.

Q: Give equations for two distinct planes whose trace in the yz-plane has equation y = 4z

A: Given, two distinct planes have trace in the yz-plane has equation y = 4z

Q: 3. Show the parametric form of the line that passes through points A(3, 6, 1) and B(2,10, 5) Compute…

A:

Q: Let L, be the line through the points (1,2,6) and (2,4,8). And let L2 be the line of intersection of…

A:

Q: Calculate | F.dr, where F(x, y) = (x²–3y)i+(ye")j and the curve and curve C is the parabola æ = y² +…

A: We have to calculate ∫CF.dr, where Fx,y=x2-3yi+yeyj and the curve C is the parabola x=y2+1 from A1,0…

Q: 1) Find the distance from the point (1, 2, 3) (a) XZ plane to the (b) Z axis

A: In this question we will use basic of geometry i.e concept of 3D geometry. We will first plot the…