1. Let S be the surface generated by the curve +y+4= 0 when it is revolved about the z-axis. (a) Give a set of parametric equations for S. * (1. 4. 4). (b) Use the answer in (a) to find the normal line to S at

Q: Find the parametric equations of normal line to the surface given by f(x,y) = tan Vx2 + y² at the…

A:

Q: 4. Find equations of the tangent plane and the normal line to the surface yz = arctan(2x+z) at the…

A:

Q: Find parametric equations for the normal line to the surface z=-7x2 -9y2   at the point   (2,1,-37)

A:

Q: 6) Find the equation of the osculating plane of the helix x = sin 3t, y = 2t, z = cos 3t at the…

A:

Q: Find parametric equations of the tangent line to the curve of intersection of paraboloid z = x² + y²…

A: We know that tangent line to ellipse will be perpendicular to normal of both the surface. Therefore,…

Q: 4. Find the parametric equations for the tangent line to the curve of intersection of the y? z1…

A:

Q: Find parametric equations of the tangent line to the curve of intersectic paraboloid z = x² +y? and…

A:

Q: 4. Find an equation of the tangent plane to the surface z = xy+8 at the point (1,1,-10) xy-2x

A:

Q: 15. Find the radius of curvature at 0 = -n of the parametric equations x = 3(0 + cos 0), y = 3(1 –…

A: For the parametric curve x=x(θ), y=y(θ)Curvature K=|x'y''-y'x''|[(x')2+(y')2]32Radius of…

Q: find equations for the(a) tangent plane and(b) normal line at the point P0 on the given surface. cos…

A: Given,

Q: Find an equation for the tangent line to the curve given parametrically by x(t) 2t, y(t) 4t2 +t+ 3 =…

A: we want to find the tangent lines to theparametric equations given byx=f(t)  y=g(t)The equation of…

Q: 15. Find the radius of curvature at 0 = ÷n of the parametric equations x = 3(0 + cos 0), y = 3(1 —…

A: Given two parametric equations, x=3(θ+cosθ),y=3(1-cosθ) To find the radius of curvature at θ=π3

Q: A Cartesian equation for the parametric curve x=24sin?e, y=21cos20, n ses 31/2 is given by O…

A:

Q: (7.5.0)- Find an equation of the tangent plane to the surface of z = sin(x – 2y) at the point %3D

A: Given F(x, y, z)=z-sin(x-2y) ∂∂xF(x, y, z)=∂∂x(z-sin(x-2y))=-cos(x-2y)∂∂yF(x, y,…

Q: Find an equation of the normal line to the surface given implicitly by z = sin(xy) at the point…

A:

Q: Find an equation of the tangent plane to the surface at the given point z= 5x - xy3+y2,     (1,1,5)

A:

Q: z = In(x²y*) + / x² + y²

A:

Q: Find parametric equations of the normal line to the surface z = + y? at the point (2, -2,5) x = 2 -…

A: Let F(x,y,z) define a surface that is differentiable at (x0,y0,z0) then the normal line to F(x,y,z)…

Q: #5. Find an equation of the tangent plane to the surface with the equation 12x z = x²y*. y at the…

A: The given equation of the surface: z=x2y4-12xy Therefore: f(x, y)=x2y4-12xy Now:…

Q: Find an equation of the tangent plane to the surface z = 16 − x2 − y2 at the given point (2, 2, 8)…

A: To Determine: Find an equation of the tangent plane to the surface z = 16 − x2 − y2 at the given…

Q: Find parametric equations for the normal line to the surface x2 + 9xyz + y² = 11z2 at the point (1,…

A: Given:- x2+9xyz+y2 =11z2  To find:- Equation of normal line to the given surface at point (1,1,1)

Q: Determine the equation of the normal line to the surface z2 cos(x + 2y) at the point (2, –1, 0).

A:

Q: Find equations for the a) tangent line and b) normal plane to the curve made by the intersection of…

A: Given The given surfaces are 3x2y+y2z=-2 and 2xz-x2y=3.  The given point is 1,-1,1.

Q: Compute the length of the curve given parametrically by x=1/3t^3 and y=1/2t^2 for 0 less or equal to…

A:

Q: 4. Find the parametric equation of the tangent line to r(t) = 4costi – 3tj %3D at Po(2, –n)

A: Given: To find the parametric equation of the given tangent line as follows, r(t) = 4 costi- 3tj…

Q: 9. Find the exact length of the parametric curve for the given interval. x = 4t 2 - 3, y = 3t, 0 St5…

A:

Q: 1. Eliminate the parameter to find a Cartesian equation of the curve given parametrically by (x(t),…

A: Recall: sin2t+cos2t=1

Q: Obtain the gradient to a curve with the parametric equations given by I = In - and y = sinh 2t whent…

A:

Q: 4. Find an equation of the tangent plane to the surface z=- xy+8 at the point (1,1,-10) ху — 2х

A: Equation of tangent plane is: {df(x0,y0)/dx} (x-x0)+{df(x0,y0)/dy} (y-y0)-(z-z0)=0

Q: 10. Find the value of 2 for the parametric curve xx= y = 2t2 -6 at the point (1.-6).

A:

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A:

Q: 4. Find an equation for the tangent plane to the surface z = 3x + 5y + arctan in the point (0, 1,…

A:

Q: 4. Find the parametric equations for the tangent line to the curve of intersection of the y z 1…

A: Given:

Q: 4. Find the equation of the tangent plane to the surface ey + In(xz) – evz = 0 at the point (1,0,…

A: Given:

Q: Use definition 13.7.1 to find the equation of the tangent plane and the parametric equations for the…

A: Let the given function is,

Q: Find the points (x, y) on the curve with the parametric equations (x3Dt3- 4t at ly=t +2 which the…

A:

Q: 9. Find an equation for the tangent plane and the normal line to the surface xy + yz + zx = 5 at the…

A: f(x,y,z)=xy+yz+zx-5=0,  at  (1,2,1)now we differentiate above equation with respect to x,y,z…

Q: Find the parametric equations of normal line to the surface given by f(x, y) = tan v x2 +y2 at the…

A:

Q: Find the slope of then tangent line to the parametric curve t t2 + 25, y 6. at t = -1 and t = 1…

A: Shown below

Q: 4) Find the critical points of the parametric equations x = 5 – In t y = t + Int

A:

Q: 3. Find an equation of the tangent plane at P = (0, 3, –1) to the surface with equation ze" + elz+1)…

A:

Q: Eliminate the parameter in x = -t +2t and y =t-4 and then identify the parametric curve and sketch…

A:

Q: 4. Let S1 : x + y – 2z +1 = 0 and S2 : x² + y – z = 0 be two surfaces. Find symmetric equations of…

A:

Q: 2. Find a set of parametric equations for a particle that oscillates on the line y = 6x – 5 between…

A:

Q: Find an equation of the tangent plane to the given surface at the specified point. 3.  z = 2x2 + y2…

A: Given that: z = 2x2 + y2 - 5y, (1, 2, -4)

Q: Find the surface of revolution of the curve y = 4 + 3x² that is rotated about the y- axis between 1…

A: We have to find the area of the surface of revolution of the curve y=4+3x2 that is rotated about the…

Q: Find an equation of the tangent plane to the surface z = √(9 − x2 − y2), at the given point (1, 2,…

A: The surface given is Fx,y,z=√(9 -x2 -y2) -z =0. The derivative of the surface coordinates with…

Q: 1. Find the equations of the normal and osculating planes of the curve x Int, y 2t, z = t2 at the…

A: Equation of the plane :  r→.n^ =a→.n^ ; where 'r→' is the position vector and 'a→' is the given…

Q: Find an equation of the tangent plane to the surface x2 + y2 + z2 = 9 at the given point (1, 2, 2)…

A: Given equation: x2+y2+z2=9 Consider the given equation, fx,y,z=x2+y2+z2-9=0

Q: Determine the equation of the normal line to the surface z2 = cos(x + 2y) at the point (2, –1,0).

A: We have  z2 = cos(x+2y) ⇒cos(x+2y) - z2 = 0  So, F(x,y,z) = cos(x+2y) - z2  Thus,  ∇F(x,y,z) =…

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.