Let G(u, v) (2u + v, 9u + 2v) be a map from the uv-plane to the xy-plane. Describe the image of the line through the points (u, v) - (1, 1) and (u, v) - (1, -1) under G in slope-intercept form. y =

Q: Find a function r(t) that describes the line through P(3,4,5) that is orthogonal to the plane 2x-z=4

A:

Q: Find the mapping that reflects the plane in the line x+y-1=C O (x,y)>(y+1,x+1) (x,y)→(-x+1,-y+1) O…

A:

Q: find the intersection point between the line      X=1+2t {    y=2-t       Z=3 and the x+y-z=3…

A: The intersection point will be common to both the line and the plane. Therefore, substitute the…

Q: Calculate the equation of the line that passes through the point (0, 1, 2) that is parallel to the…

A: We need to find the equation of the line passing through a point, parallel to a plane and…

Q: Find the tangent plane for the level surface f(x,y,z) = 5x3 + cos (5y) + In (6z) = c at the point…

A:

Q: Find the tangent plane (Xo, Yo, Zo) from f (x, y) = x² - 3y²; (0,1, -3)

A:

Q: Let F(x, y) = (cos(x*) -v°)i + x*j. F• dr, where C is a curve consisting of the line segment joining…

A:

Q: Describe the image of the line z = 1 + iy under the map flz) = sin z %3D

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: find the trace of the plane in the given coordinate plane.  −x + y = 4, xz

A: Find trace of the plane in the coordinate plane -x+y=4 in xz

Q: Let f(x,y) = x* + y². Find the slope of the line tangent to this surface at the point (1,1) and…

A:

Q: Determine the location of a point (x, y, z) that satisfies the condition y < 0

A: With the given condition there are 4 scenarios 1) x>0,y<0,z>0 2) x<0,y<0,z<0 3)…

Q: Find the distance between the planes x + 3y – 2z = 3 and 2x - y - z = 4. %3D

A:

Q: Find the distance from the point (2, 0, 0) to the plane x + 2y + 2z = 0.

A: Given a point, (2, 0, 0) and the equation of the plane x + 2y + 2z = 0.

Q: Find the equation oftangent plane and normal lines ofthe surface: f(x,y, z) = 10x² –14 y² +11z³-7 at…

A:

Q: find the trace of the plane in the given coordinate plane.  −x + y = 4, yz

A: Let the plane P whose equation is;                                      -x+y=4…

Q: find the gradient of function f [x, y,z] = 3x² -y?_z? at paint {i,-Ź, -¡}

A: The given function is fx,y,z=3x2-y3-z2 and the given point is 1,-2,-1. The gradient of the function…

Q: Show that the line x + 1 =y/−1=(z − 2)/2 is in the plane 2x + 4y + z = 0.

A: Given equation of a line, x+1=y-1=z-22 And equation of a plane, 2x+4y+z=0

Q: Find a harmonic function ¢(x, y) in the right-half plane x>-1 such that on the boundary (-1, y) = 0,…

A:

Q: Find the plane containing the intersecting lines x = -1+t, y = 2+ t, z =1–t; -0 <t < o∞ | L2: x = 1…

A: Find the planes containing the intersecting lines: L1:x=-1+t,y=2+t,z=1-t for all…

Q: Example: find the tangent plane and normal line of the surface f(x, y, z) = x + y' +z-9=0 at the…

A:

Q: ▪ Let L:2 y+3 Z+4 X-2 = = and M: 2x - 3y + z = 6 are line and 2 5 plane and let P be the…

A: First we find the intersection point of line and panel

Q: What is the symmetric equation of the line that is normal to the plane x + y + z = 9 at the point…

A:

Q: Find the slope of the surface z = f (x, y) in the y-direction at the point (64, 8).

A: Follow the procedure given below.

Q: Find the point in which the line through the (3,-1,2) perpendicular to the plane x +2y -5 z = 4…

A:

Q: Sketch the surface of f(x,y)

A: To sketch the function

Q: Show that the line through two distinct points (x1, y1) and (x2, y2) in the plane has equation: | x…

A: Solution..

Q: Let f(x, y, z) = 3xy + 2yz - xz. Find the directional derivative of f(x, y, z) in the direction of…

A:

Q: Find the distance between the point (3, 1, −2) and the plane x + 2y − 2z = 4.

A: Given,             Point  (3, 1, −2)          Plane   x + 2y − 2z = 4

Q: Find the distance from the point (1, 1, 1) to the plane x + 2y + 3z  = -4

A:

Q: Example: find the tangent plane and normal line of the surface f(x, y, z) = x + y° +z-9=0 at the…

A:

Q: Select all lines that are parallel to the plane x+y+z%= 3. OF(t) = (1,1, 1) + t(1, 1, –2) OF(t) =…

A:

Q: Find √x²ds ds along the line of intersection of the two planes x-y+z=0 and x+y+2z=0, from origin to…

A: The two given planes are x-y+z=0 and x+y+2z=0.

Q: Find the distance between the point (3,11,-2) and the plane X t 2y

A: Explanation of the answer is as follows

Q: Find the slope of the surface z = 3x/y at the point (5,2,15/2) in the x- and y- directions.

A: Given data is: z=3xy  To find the slope of the surface z at the point 5,2,152 in the x and…

Q: Let ƒ(x, y) = x2 + y3. Find the slope of the line tangent to this surface at the point (-1, 1) and…

A:

Q: Find an equation of the tangent plane to the surface x-xyz 56 at the point (-4,5,2). Select one:…

A:

Q: Evaluate //R V2x(y – 2x)dA where R parallelogram on the xy-plane with vertices (0, 0), (0, 1), (2,…

A:

Q: Let ƒ(x, y) = 2x + 3y - 4. Find the slope of the line tangent to this surface at the point (2, -1)…

A:

Q: Find the distance between the plane x+y +z = 3 and the line L= λ(1,0,-1).

A:

Q: Find the point on the plane x+y+z%3D24 that is closest to the point (1,1,1).

A:

Q: Select all lines that are parallel to the plane x+ y+ z = 3. OF(t) = T (t) (1, 1, 1) +t(1,1, –2)…

A: We are given a plane x+y+z=3. We have to determine which lines are parallel to the given plane among…

Q: Let f(x.y) = x° +y*. Find the slope of the line tangent to this surface at the point (- 1, - 1) and…

A:

Q: Find the distance between the point Q (3, –2, 4) and the plane 2x – 5y + z = 10 -

A:

Q: Find the plane determined by the intersecting lines. y = 2+ 4t L1 x= - 1+2t z = 1- 2t L2 x= 1-4s y =…

A:

Q: Find the angle between the line 2 x+v3 : = z + 1 and the plane vZx + y + z – 2 = 0.

A:

Q: Let z=f(x,y) = y x². Find the slope of the line tangent to this surface at the point (-2, -1) and…

A: 5.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.