Prove that at the point of intersection of the surfaces x² + y² = z², z = a_tan` (*) vhere y = x tan 0, the radius of curvature and torsion are given by a(2+0²) ³/2 (8+50+0¹) ¹/2 p = and o = a (8+50² +0¹) 6+0²

Q: ) Find symmetric equations for the tangent line to the graph of f(r, y) = at (1, 2, 2) in the plane…

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Q: 3. (a) Show that the two surfaces S1 : z = xy and S2 : z =3x^2/4 - y^2 perpendicularly at the point…

A: As per our guideline, we are supposed to solve only first subpart. Kindly repost other subpart as…

Q: Exercise 8. Show that the sum of the angles of a geodesic triangle on a surface of positive…

A: We will use the Gauss–Bonnet theorem to prove the given statement.

Q: Q9)The curvature of the circle * x^2+ y^2=9 is equal to 1/2 F O

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Q: ind the parametrization of the curve defined by the intersection of the two surfaces y = x^2 − 3z^2…

A: To find the parametrization of the curve defined by the intersection of the two surfaces  and ,…

Q: Calculate the curvature of r (t) = 〈t,t^2,t^3 〉 at the point (1,1,1)

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Q: Prove that at the point of intersection of the surfaces x² + y² = 2², z = a tan` X where y = x tan…

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Q: where y . Prove that at the point of intersection of the surfaces (4) y __x² + y² = z², z = a tan` =…

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Q: Determine the curvature of the curve x=y^3 at point (1, 1). a) 10.89 b)11.12 c) 8.98 d) 12.2

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Q: PCt) =<5-t, 4t -3,3t7 uking are length Reparami tize parameter from the point (4,113).

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Q: An equation for the tangent plane to the surface (xty at the point P(1, 1, 4) is a) 8r + 8y-:= 0 b)…

A: The given surface is  f(x,y)=z=(x2+y2)2  ...........(1); we have to find the tangent plane at point…

Q: If a parametrized curve is given by: r⃗ (t)=⟨2t^3,10ln(t),5t^2+9t⟩ At t==2, the position vector is…

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Q: Find the lines of curvature of a surface x=cos⁡u y=sin⁡v z=v

A: Given: Surface: x=cos u, y=sin v, z=vTo find: The lines of curvature of the given surface.

Q: 4) Find the families of lines of curvature of the surface z = r + y? .

A: Given: z=x2+y2

Q: Let T: P2 P3 be the linear transformation defined by .3x T(p(x)) = (x + 1)p(x) + p′(1) + Find [7],…

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Q: True or false: the curvature of,k, for x^2+y^2=100 is larger than for x^2+y^2=1/4

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Q: Consider a plane curve defined by the parametric equations :  x(t)=acos^3 (t) and  y(t)=asin^3 (t)…

A: The curvature κ(t) of a plane curve defined by parametric equations x(t) and y(t) is given…

Q: Find the equation of the tangent plane to the given surface at the specified point   2x=y^2+z^2-9 at…

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Q: Find the angle at which the circles x² + y² + x + y = 0 and x² + y² + x − y = 0 ir If the circles x²…

A: The angle of intersection of the two circles is given by cosθ=r12+r22-C1C22r1r2where r1 and r2 are…

Q: Let C be the curve of intersection of the surfaces y = V2 x2 and z 2/2 ху. (a) Compute the curvature…

A: Given C be the curve of intersection of the surfaces y=2x2 and z=223xy

Q: True or False (iv) The line joining the Sun to a planet sweeps out equal areas in equal times. (v)…

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Q: Consider the curve given by r(t) = where t varies over the real numbers. Find the curvature of r(t)…

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Q: drawn from the point A(2,10) on the parabola y = x^2 + bx + c Since the tangent is parallel to y =…

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Q: Give an arclength parametrization of the line which is the intersection of the tangent planes of…

A: Let First, find the tangent planes at (1, -1, 0), (1, 2, 9).     Tangent plane at (1, -1, 0):…

Q: find the Cerrvature of the Curve x:(0,1|R? defined by x(t)=14Cost, 1sint,o), where %3D is coostant 2

A: α(t)=(1rcost ,1rsint ,0)r is constantTo find the curvature of the given curve