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

Q: Show that the square of the length of subtangent at any point on the curve by2=(x+a)2 (b is not…

A: Here given curve  by2=x+a2

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

A:

Q: Suppose that ∇fp =〈2, −4, 4〉. Is f increasing or decreasing at P in the direction v =〈2, 1, 3〉?

A: 1) v=2,1,3 v=22+12+33 = 14 u=vv=1142,1,3

Q: State fuo oer femnme egual JF.T ds gimn fow line into f.T ds givern that

A: In this question we have to find two other form of line integral.

Q: Evaluate the line integral ∫_c (2x − y) dx − (x+ 3y) dy , where C is a straight line from (1,1) to…

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Q: Wnete to y=2x=1 equotion for Hhe line Pacallel the Doinit 4,9) an and

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Q: The direction field of the DE y' + ye³x+7 = −2y² will have a line segment with positive slope at the…

A: Given y'+ye^(3x+7)=-2y² We know that the dy/dx gives the direction field. Hence the given DE can be…

Q: The line that passes through the points (2,0), (0,1) is x+2y = c, where c = 1. O True False

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Q: Now compute [(x − 3y² + 2) ds where C' is the union of the curves C1 and C2. C1 is the line segment…

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Q: Let Ē = Calculate F. dr where r is the Straight line seg ment connecting (1,1) to (2,2)

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Q: 1. Determine whether the lines L₁ given by z=t, y=1 with -∞ <t<∞o) and L₂ given by z=2+s, y = 3-2…

A: Introduction: If two lines are parallel then the ratio equality is true between them. And if two…

Q: 13. Show that the length of the portion of any tangent line to the ball x2/3 + y2/3 cut off by the…

A: Given: The equation of the ball is: x23+y23=a23...........(1) We have to show that the length of the…

Q: Let f(x,y) = ax + by2 af where a and b are constants. If of at x = 1 and %3D ху y 2, then the…

A: Introduction :

Q: If (x, y) > (x+3, y-7) maps TUVW to T'U'V'W', what is the algebraic description that will map…

A: x, y→x+3, y-7 maps TUVW to T'U'V'W'. That is, the x-coordinates of T, U, V and W are added by 3…

Q: If C is the line segment which begins at (-2, 1) and ends at (2, 4), evaluate the line integral: et…

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Q: Show that if y is the boundary

A: To prove that if γ is the boundary of the trianglee with vertices at z=0, z=3i,and z=-4 oriented in…

Q: Evaluate ſc(x – y) ds over the path C: F(t) = 4tỉ + 3tj for 0 <t< 2.

A: Given: The curve C: r→t=4ti→+3tj→ for 0≤t≤2 To Evaluate: The integral∫Cx-yds

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: Compute x²z ds where Cis the line segment from (0,6,–1) to (4,1,5).

A:

Q: 15. When a line L is tangent to the circle 2 + v = 100 at the point (a, 6), find the equation of…

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Q: C is the line segment from (-1,0, 3) to (1, 3, 4), and F(x, y, z) = zi+xj+yk, evaluate Jo F.dr.

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Q: Find the distance from the point (1,1,5) to the line X=1+t, Y=3-t, Z=2t. O (5)^0.5 O 7 O 2(3)^0.5 O…

A: Given: Equation of line is: X = 1+ t Y = 3 - t Z = 2t To find: distance from the point (1, 1, 5) to…

Q: Compute the length of the path r(t)= (t, 4t3/2, 2t3/2) over the interval 0<t<8.

A:

Q: ompute the line integral ∫C [2x3y2 dx + x4y dy] where C is the path that travels first from (1, 0)…

A: The line integral is given ∫C2x3y2dx+x4y dy Consider fx,y=x4y22. ddxx4y22=12·4x3y2+22x4y…

Q: 1. Evaluate 4x³ds, where C ie the line segment trom -2,-1,QL,4). らt Xz xlt)= (1-t)(-2) + t(1) E) :…

A: Introduction: A line integral is an integral that determines the function to be integrated along a…

Q: 3e) if c is any Simple Closed Curve, evaluate f dz, if ü7is out side (ii) 7 is in side

A: We will solve it by Cauchy Residue Theorem ∫f(z) dz = 2πi×R where R is the Residue of the…

Q: The tangent to the circle x2 +y2 +10 x+5 y= 60 at the point (2 ,4) is Select one: O None O 13 y +…

A: Given that:x2+y2+10x+5y=60

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

A:

Q: 5) s how that the line of intersection f the Plas and 5x-2y-Z=0 is Parrliel to theline X+2y-22= 5…

A: Given that plane  ,P1:x+2y-2z=5plane , p2:5x-2y-z=0line ,l:x+32=y3=z-14Normal of P1  ,…

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

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Q: Let C be the line segment connecting 0 to ni. Then f. 1-e2z dz =

A: we have to find integration

Q: Let y = [3] and u = [2] to the line through u and the origin. Compute the distance from y

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Q: | 6. Let y = 3 and u = . Compute the distance from y to the line through u 1 2 and the origin.

A: Given: y=-231  and u=1-12 To compute: the distance from y to the line through u and the origin.…

Q: 4. Prove that the following curves are intersecting with a right angle: 5y-2x+y²-x²y = 0 2y +…

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Q: Find a distance from the point S(1,1,5) to the line x=1+t, y=3-t,and z= 2t

A: We need to find the distance of the point (1,1,5) and the given line.

Q: Compute (x² + 2y + 3z +1) ds where Cis the line segment from (0, 2, 3) to (2, 5, 7).

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Q: and u = then the distance from y to Span{u} = line through the origin and y is 2/10. Let y = True…

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Q: Evaluate the line integral 3ydx + 6xdy where C is the straight line path from (1, 2) to (6, 5). Sc…

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

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Q: 6 - Find the length of the curve y = (x)7 from the x = 0 to x = 4. a) O 8/27(10^(3/2)+2) b) O…

A: .

Q: Let s(xy) = x²+y²_2x – 16y +10 =0 a cirdle be L (xy) = X-5y + 10=0 be a line.

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Q: The area bounded by the curve y = 6z and the lines a = 5 and z= 8 above the T aris is 234.00 square…

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Q: Evaluate the line integral ∫Cx2yds along the segment of the line y=x from the origin up to the point…

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Q: Find the distance from the point (1,1,5) to the line X=1+t, Y=3-t, Z=2t. *

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Q: .11 Compute (1/ry, 1/(x + y)) · dr along the curve (2t, 5t), 1 < t < 4.

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Q: O whion of the line: y--2x+8 Poliowing line5 ave perpendicular to this. B. ut

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Q: 5. Find the distance between the point (4, 3, 1) and the line f(t) = (2x - 1, –3y + 1, 2)

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Q: 5. Show that the following orthogonal families of curves: are x² + y² : = ax, x² + y? = by

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Q: Evaluate | xyzdx + In(xy)dy + xzdz over the straight line segment from (2, 1, 1) to (–1,1, –2)

A: We need to evaluate integral∫Cxyzdx+lnxydy+xzdz over the line segment from 2,1,1 to -1,1,-2.