Project: S is a surface in R° with equation x² + y² – z² = 1. The point P (2,1,2) lies on this hyperboloid. Itturns out there are exactly 2 straight lines L, and L2 which pass through the point P and lie entirely inside thesurface S.Your job: Find the parametric equations of these 2 lines and then display the surface S together with the lines L1and L2 using computer graphics software.Methods: Theory: For finding the equations, I suggest this approach. (You can use a different approach if you knowa better one.) The two lines can be found if you know direction vectors for them. To get the direction vector v =(a, b, c)for L1 or L2 write down the parametric equations of a line with direction vector v and passing through P.The condition that the line lies on S gives an equation involving t, a, b and c. Solve this equation to find twosolutions for a,b,c.

Clearly : x-z = 0 and y+1 = 2. So, we can write: 1.(x-z) = 0.(y+1). So, L1 can be written…

Project: S is a surface in R° with equation x² +y² – z² = 1. The point P (2,1,2) lies on this hyperboloid. It turns out there are exactly 2 straight lines L1 and L2 which pass through the point P and lie entirely inside the surface S.

Project: S is a surface in R° with equation x² +y² – z² = 1. The point P (2,1,2) lies on this hyperboloid. It turns out there are exactly 2 straight lines L1 and L2 which pass through the point P and lie entirely inside the surface S.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 53E

See similar textbooks

Related questions

Q: Calculate | [(a4 – 3y) da + (2y³ + 4.x) dy] wherein r(t) = (3cos(t), 3sin(t)), 0 <t < 27, is an…

Q: State the type of the quadratic surface: ()' - (5)' + (¿)° = 1 2 1. Ellipsoid 2. Hyperboloid of one…

A: Here we need to find the type of quadratic surface and it's trace obtained by the plane z=1

Q: The quadric surface, * - a - = 1 l is a hyperboloid %3D 64 None of these of one sheet that opens…

A: The given quadratic surface is: x24-y264-z216=1

Q: The plane x+ y+2z =2 intersects the paraboloid z = x2+y² in an ellipse. Find the points on this…

A: Given a plane x+y+2z=2 intersects the paraboloid z=x2+y2 in an ellipse We need to find the points on…

Q: 2. Which equation represents the parabola with focus (1, 4) and latus rectum of 4? Oy = 4(a- 1)2 + 3…

A: Here, the parabola is of the form (x-h)2=4a(y-k) Whose focus is given by (h,k+a)=(1,4) given…

Q: %3D at

A: We need to substitute x=au ,y = bv ,z= cw

Q: The plane 15x + 20y + 40z = 195 intersects the cone z = x² + y² in an ellipse. The sum of the…

Q: Find the equation of the xy-trace of the sphere. (x – 2)? + (y + 4)² + (z – 6)? = 72 Sketch the…

Q: Find an equation for the surface obtained by rotating the parabola y=2 about the y-axis.

Q: z = 9x? + 9y? O b. z= 3x2 + 3y? O c. z= 4x? + 4y? O d. z= x2 + y? z = (x – 3)2 + y² Oc. O e.

Q: State the type of the quadratic surface: (중)"- (1)" + (주)'=1 1. Ellipsoid 2. Hyperboloid of one…

A: NOTE: Refresh your page if you can't see any equations. . here two of the variables have positive…

Q: z' +z+1 Evaluate -dz , where C is the ellipse 4x + 9y = 1. c z² – 3z+2

A: By Cauchy integral theorem, If the function fz in analytic in some simply connected…

Q: Find the equation of the surface of revolution when the hyperbola (y^2 / 9) − (z^2 / 4) = 1 is…

A: Find the equation of the surface of revolution when the hyperbola (y^2 / 9) − (z^2 / 4) = 1 is…

Q: Identify the surface 2x² + 4y? – 1=0. A. An elliptic cylinder. B.Ahyperbolic cylinder. OC. None of…

A: We will solve the problem

Q: A] Find the length of the arc of the semi-cubical parabola y² = x3 (1,1) and (5,5V5). between the…

Q: • Which of the following conics is a trace on one of the coordinate planes of the surface whose…

Q: 6. In Section 12.5, you learned that the distance from a point (ro, Yo, zo) to a plane ax+by+cz = d…

Q: 2. Find a parameterization of the line formed by the intersection of the following planes: P1: 2x –…

A: Line of intersection

Q: What is the highest point on the ellipse that is the intersection of the plane z+y+ 2z=2 and the…

Q: The plane 6x + 8y + 60z = 700 intersects the cone z = x + y in an ellipse. The sum of the z-…

Q: - The surface 5x^2 + 2z^2 = 3y is heading towards which axis? - Identify the surface generated by…

A: 5x^2 + 2z^2 = 3y x= 3n1 , y = -3(5n12 + 2n22) , z = 3n2

Q: Given z=-70-20x2 is a curve in the xz-plane. If the equation of the surface generated by revolving…

Q: Match the quadric surfaces to its graph. A. y + = 4+ 4x² C. 9z + 4y = 36 + 36z² B. 42 + y? + 4z² =…

A: We have to identify which Quadratic equation corresponds to which of the given 3D graphs.

Q: F. dr, where C is the ellipse (x + 3)² + 5(y – 2)² = 16 and F(r, y, z) = x²i +y²j+ z²k.

A: The vector field is Fx, y, z=x2i+y2j+z2k. The curve C is x+32+5y-22=16. Obtain the parametric…

Q: Let S be the quadric with equation 36x² – 4y? – 9z2 – 36x = 0 - - (a) Classify the graph of S. (b)…

Q: (b) Find the equation of the tangent plane to the hyperboloid r2 + 3y? – 322 = 1 at the point (1,…

A: We have to find the equation of the tangent plane to the hyperboloid x² + 3y² − 3z² = 1 at the point…

Q: Prove that evolute of the ellipsé x2 y? = 1 a? b2 is (ax)2/3 +(by)2/3 = (a² – b²)²/3. %3D

A: Given equation of the ellipse be x2/a2 + y2/b2 = 1 Write the equation of the ellipse in parametric…

Q: 4) Consider the images of the hyperbolas r - y = ±c? under the mapping f(2) = 22, where c > 0. Show…

Q: Identify the surface that corresponds to the equation 100y^2 - 25x^2- 4z^2 = 1. O elliptic…

A: To identify the surface that corresponds to given equation- 100y2-25x2-4z2=1.

Q: Find the coordinates of the four points at which the hyperboloid of one sheet given by x2 y? + 16 z?…

A: This question is related to 3D coordinate geometry, We will solve it using given information.

Q: The surface defined by the equation z= 4x2 + y2 is called an elliptical paraboloid. a. Write the…

A: Given, function

Q: F. dr, where C is the ellipse (x + 3)² + 5(y – 2)² = 16 and F(x, y, z) = x²i+ %3D Pj+ z°k.

A: Given: x+32+5y-22=16 Fx,y,z=x2+y2+z2

Q: x + y? – 4y + 2² = 0

A: Consider the provided question, Given equation of the surface, x+y2-4y+z2=0

Q: The plane x + y + 2z = 2 intersects the paraboloid z = x² + y² in an ellipse. What is the point on…

A: We will use Lagranges multiplier method to find optimal solution To maximize/ minimize f(x,y,z)…

Q: Find the equation of the surface of revolution that is generated by rotating the parabola y = x^2…

A: Equation of surface of revolution of a curve about y-axis is given by x2+z2=r2 where r is the radius…

Q: For the quadric surface, - + z = = 1, setting the value of z = 1will give you which conic section? 4…

A: The given quadric surface is x24-y29+z=1 To find when z=1which conic section represents the…

Q: Find an equation of the parabola deseribed. Focus at (5, 0); vertex at (0, 0) O y = 20x2 O x = 20y O…

Q: Find the equation of the surface of revolution generated by rotating the parabola y = z^2 around the…

A: A surface of revolution generated by rotating a curve described by y=f(z) around z-axis is given as…

Q: The plane x + y + 2z = 6 intersects the paraboloid z = x2 + y² in an ellipse. Find the points on…

A: To find the farthest and nearest point from the origin to the ellipse.

Q: Consider the following equations of quadric surfaces: b) x² – y² + z? = 4 i) Identify the type of…

A: Given equation of quadratic surfaces: x2-y2+z2=4

Q: Consider the ellipse given by the intersection of the plane x+y+7z=3 with the paraboloid x^2+y^2=2z.…

Q: By differentiating implicitly, find the slope of the hyperboloid 22 + y? – 22 = 1 in the x-direction…

A: We have to find slope of the hyperboloid x2+y2-z2=1 in the x-direction at the points (a) 1,2,2 (b)…

Q: Consider the following equation of a quadric surface. y? 2 2 X + 144 144 1 144 - a. Find the…

Q: Consider the following equation of a quadric surface. 9x2 + 4y? + 4z? = a. Find the intercepts with…

A: Question is solved.

Q: 15) Consider the following equations of quadric surfaces: a) x = + - 4 b) x? - y? + z? = 4 i)…

A: We need to find surface type and intercepts.

Q: Find equetion of Hhe parabole farmm an I with focus zt (3,0) and divectrix x=-3

A: Focus = F= (3,0) Directrix - x =-3 i.e. x+3=0

Q: Given the surface of revolution with equation 22 y? 3x. 3 3 Find a generating curve of the surface…

Q: What is the DE of the family of parabolas having their vertices at the origin and their foci on the…

A: Make equation of parabola

Q: If h, k,1, a, b € R+ . Of the quadric surfaces whose equations are shown below, which one generates…

A: Given: z=l+1 The standard form of the equation of a hyperbola with center h,k and…

Q: Describe the graph of the equation. r = 9cos ti – 19sin t j + k O An ellipse in the plane z = 1,…

Question

Project: S is a surface in R° with equation x² + y² – z² = 1. The point P (2,1,2) lies on this hyperboloid. It
turns out there are exactly 2 straight lines L, and L2 which pass through the point P and lie entirely inside the
surface S.
Your job: Find the parametric equations of these 2 lines and then display the surface S together with the lines L1
and L2 using computer graphics software.
Methods: Theory: For finding the equations, I suggest this approach. (You can use a different approach if you know
a better one.) The two lines can be found if you know direction vectors for them. To get the direction vector v =
(a, b, c)for L1 or L2 write down the parametric equations of a line with direction vector v and passing through P.
The condition that the line lies on S gives an equation involving t, a, b and c. Solve this equation to find two
solutions for a,b,c.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 7 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Hyperbolas$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.