show that k and t are both zero for the line r(t) = (x0 + At)i + ( y0 + Bt)j + (z0 + Ct)k.

Q: Q7/ Calculate , F.dr, where F=xi+xyi and e is the boundary of the square in the plane z-0 and…

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Q: Evaluate the line integral. | (9x – 9y) ds C C : x² + y² = 64 | (9x – 9y) ds = C

A: Explanation of the solution is given below...

Q: Let f(z) = x² +iy² where z and y are the real and imaginary parts, respectively of z and C is a…

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Q: show.. that the line in.which. the planes.X+zy-223D5 and. 5X-zy.-7=D0.jntersect.is. pavalle.to. the…

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Q: Sketch in the two level curves of: f (x, y) = y+ x² = k _for k = 1, 2.

A: Solution is written in below steps.

Q: Find the point at which the line f(z)= 5z + 21 intersects the line g(z) = - 2z -7 %3D -2 11 Submit…

A: To find the intersection point of the lines f(x)=5x+21 and g(x)=-2x-7.

Q: Find the equation of the tangent line to the given curved as directed. x²+2y=8 perpendicular to the…

A: topic - equation of tangent line

Q: 2. Let j = Find the point on the line y x that is closest to j. %3D %3D 26

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Q: Find the equation of the line tangent to Y 3x2 + x + 1 at x = -3. у 3 —17г — 26 O y = 6x – 3 O y =…

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Q: +3そ20 22 Prove that the line X21+2t ニー+3t is perpendicular to the plane X-2y+zz6

A: As per our guideline we are allowed to do 1 question at a time. Please post other questions next…

Q: (5xy --- 6x(2^

A: Given,        F=(5xy-6x2)i+(2y-4x)j and   y=x3 from the point (1, 1) to (2, 8).

Q: Find the point P where the line x = 1 + t, y = 2t, z = -3t intersects the plane x + y - z = 0.

A: Given Line is x=1+t,y=2t and z=-3t  And Given plane is x+y-z=0

Q: For the function f(x,y,z) = In (5x + 4y + 9z), find f,, fy, and f,.

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Q: At what point does the line: L: x=-1, y= 10 + 31, z= 4 +3t im plane 1:x- 2y + 5z + 16 = 0?

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Q: Where does the line x = 1 + t, y = 3−t, z = 2t intersect the cylinder x² + y² = 16?

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Q: Prove that the curves x2−y2=3 and xy=2 intersect at the right angle.

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Q: Evaluate the line integral 2ydx + 5xdy where C is the straight line path from (1, 2) to (6, 5). Г…

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Q: For the function f(x,y,z) = In (6x + 7y + 4z), find fx, fy, and f,.

A: Given: f(x,y,z)=ln(6x+7y+4z)To find the fx, fy, and fzddx (ln x)=1x

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: Compute the length of the curve r(t)= (-3t, 3t + 7,–7t + 1) over the interval 0 <t <7

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Q: Find the point at which the line f(r) = -z+1intersects the line g(z) = 4x +1 %3D

A: At point of intersection both the functions will have the same value therefore f = g

Q: Find the shortest distance, d, from the point (6, 0, -4) to the plane x + y + z = 4.

A: The shortest distance of a point from a plane is said to be along the line perpendicular to the…

Q: Find the image of the line y =-x under the mapping = f(z) =2² + 4z %3D %3D

A: We need to find the image of the line y = - x under the given mapping.

Q: Find f, and f, for f(x,y)=y In (2x+7y).

A: F(x,y) = y ln(2x + 7y) To find : fx and fy

Q: Find C so that the line y = 4x + 3 is tangent to the curve y x2 + C.

A: The given equations are y=4x+3 y=x2+c.

Q: Find c so that the line y = 4x + 3 is tangent to the curve y= x^2 + c?

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Q: Verify Green's Theorem for: f4y³dx-2x²dy where C, oriented anti-clockwise, is composed of the…

A: To find the integral ∮C4y3dx-2x2dy where C is the positively oriented closed curve x2+y2=9, y≥0 and…

Q: find the line ariz ation of the followwing function. fcx, y>=J 9z*+I6y² point (1,1). Use what this…

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Q: Let f (2) = x² + iywhere x and y are the real and imaginary parts, respectively of z and C is a…

A: The given problem is to find the value of Re[f(z2)], where given z2 = 8+7i and function f(z)=x2+iy2…

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

A: Solution..

Q: Find the are bounded by the curve y=x²-4, the lines y=0 and x=4.

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Q: 2 Show that the line y = x +3x+m for all values of m. y= mx+4 will touch or intersect the curve

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Q: Q2) find the distance between two lines L1: x= 1 +4t, y=5 - 4t, z=-1+5t L2: x=2 + 8t, y=4 - 3t, z-5…

A: Method 1:

Q: find the line of intersection of th e given planes. 4x+ y + z = 0 and 2x - y + 3z = 2

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Q: Find the shortest distance from the origin to the line of intersection of the planes 2x + y − z = 1…

A: Given: The planes: 2x+y-z=1 and x-y+z=2.

Q: Show that the point (x0 , y0) closest to the origin on the line ax+ by =c has coordinates

A: Distance between x0, y0 and origin (0, 0) is, D=x0-02+y0-02D=x02+y02D2=x02+y02f=x02+y02…

Q: Find the point at which the line y(t) z(t) = 2 + 4t -6-8t intersects the yz plane. 6-7t

A: The line x(t)=2+4t, y(t)=-6-8t, z(t)=6-7t intersect yz plane Note:    Multiple questions are given,…

Q: Show that the curves 2x2 + 3y2 = 5 and y2 = x3 are orthogonal.

A: To show that the two curves are orthogonal.

Q: Show that f(-ydx + xdy) where C is the line segment joining (x, y) and (x₂, y2).

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Q: Let f (z) = ² + iy² where x and y are the real and imaginary parts, respectively of z and C is a…

A: Real part of complex number

Q: Compute the length of the curve: f (x) = x² 2x, 0<x<2

A: Given:- f(x) =x2 -2x  To find:-  Length of curve over interval [0, 2]

Q: Find the point at which the line 7(t) +t intersects the | xz plane.

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Q: In the xy-plane, the line x + y = k, where k is a constant, is tangent to the graph of y = x^2 + 3x…

A: Given line is x + y = k, where k is a constant it is tangent to the graph of y = x^2 + 3x + 1…

Q: Consider the parabola f (x) = x² – 4x and the line g(x) = –2x – 1. Determine the %3D point(s) of…

A: Given parabola,y=x2-4x                     (1)Given line y=-2x-1                     (2)Put value of…

Q: 21. Show that k and 7 are both zero for the line r(t) = xo + At)i + (yo + Bt)j + (zo + Ct)k.

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Q: Suppose A = xy" where x, y e R". Show that ||A||2 = ||x||2||y||2-.

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Q: Is AX'Y'Z' a reflection of AXYZ across line g? g X. Z' Y X'

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Q: (a) Prove that the line 2x 3y + 26 =0 is a tangent to the curve x + y2 - 4x + 6y – 104 = 0 hence…

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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.