Show that the system of confocal and co-axial parabolas y²-4a(a+x) is self orthogonal, where a is paramet

Q: Identify the graph of the linear equation 4x + 2y + z = 8 in three dimensional space. (0,0,2) (0,…

A: We need to identify the graph of the function 4x+2y+z=8 We will do so by seeing if the points that…

Q: Graph the quadric surfaces given by the equations in Exer- cises 41–48. y2 41. x² = y? x2 42. z2 9.…

A: As per guidelines incase of multiple questions (or) multiple sub-parts asked we are supposed to…

Q: Find the scalar equation for the plane passing through the point P=(-5, -2, –4) and containing the…

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Q: Prove that the system of parabola y2= 4a(x+a) is self orthogonal

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Q: The scalar  equation of the plane passes through (-1, 3, -2), (-1, 2, -1) and (4, 1, -2) is:   a.…

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Q: Find a parametrization of the line through P = (4, 9, 8) perpendicular to the yz-plane.

A: Equation of Linethe Line through P =(x0,y0,z0) in the diresction of vector v→=(a,b,c)is defined…

Q: Find the scalar equation for the plane passing through the point P-(5, 3,-5) and containing the line…

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Q: Find an equation for the plane passing through the point P(-2, 1, 1) and parallel to the plane 3x +…

A: Solve for the equation

Q: Find the scalar equation for the plane passing through the point P=(4, -5, 3) and containing the…

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Q: prove that the system of parabolas y2=4a(x+a) is self orthogonal

A: We show that the family contain both members with slope m and slope -1/m. Hence the family will be…

Q: Find the trace of the plane 3x − 2y + 5z = 4 in the xy-plane.

A: The given equation of the plane is 3x-2y+5z=4. The trace of the plane in the xy-plane is the…

Q: Consider the quadric ax2 + y2 + 4z2 + x + 6y−4z = −a−10 where a is a parameter. Find all the values…

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Q: Find parametrizations for the line in which the planes 3r - 6y-2z = 3, and 2x+y-2z = 2 intersect.

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Q: Find an scalar equation of the plane containing the line x = 2 – 4t, y = 4 – 5t and z = 5+t and the…

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Q: A. Show that the planes (x+y- z = 1) and ( 2x - 3y + 4z = 5 ) are neither parallel nor…

A: Let's find.

Q: Find the point on the plane 3x + 4y +z = 8 that is nearest to (2,0,1).

A: Given the equation of plane:

Q: Give equations for two distinct planes whose trace in the yz-plane has equation y = 4z

A: Given, two distinct planes have trace in the yz-plane has equation y = 4z

Q: prove that the system of parabolas y² = 4a(x + a) is self orthogonal

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Q: Draw a sketch of the plane of 4x – 4y + 2z – 9 = 0. -

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Q: Find a parametric representation of the solution set of the linear equation.−3x + 4y − 2z = 1

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Q: Use the Lagrange multiplier method to find the point on the plane 4x +3y +2z = 0 with the smallest…

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Q: Find the orthogonal trajectories of the family of ellipses with centers at (0, 0) and two vertices…

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Q: 2. Show that ( 1, 6), (0, 3), (1, 2), and (2, 3) lie on a common parabola.Hint: For scratch work,…

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Q: Find the condition for the points (a,0) ,(h,k),(0,b) when ab is not equal to 0 to be collinear

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Q: Find an equation of the plane satisfying the given conditions.

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Q: A point moves in the (x, y) plane on the line 2x+ 3y −4 = 0. Where will it be when the sum of the…

A: We have to find a point from where the sum of square of its distances from 1,0 and -1,0 is smallest.

Q: 4. Find an equation for the plane in R³ spanned by the points (1,5, 1) and (1, 4, 2).

A: The two vectors are given as v1 = 151v2 = 142 Let xyz∈span{v1 , v2} Then xyz = mv1+nv2= m151+n142=…

Q: -axis. 5. Evaluate dV, where E is the solid tetrahedron with vertices at (0,0, 0), (2, 0, 0),…

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Q: 4. Find an equation for the plane in R spanned by the points (1, 5, 1) and (1,4, 2).

A: Given v1 = (1 , 5 , 1)v2 = (1 , 4 , 2) Let xyz be a point in spanv1 , v2 , so there exist m , n…

Q: Consider the following system: 2х + 3у — 2х х — 2у + 32 - 2y + 4z 1 1 %3D 2x – 2 (a) Find the…

A: The given equations are 2x + 3y - 2z = 1 x - 2y + 3z = -1 2x - 2y + 4z = -2 to find the parametric…

Q: Give equations for two distinct planes whose trace in the xy-plane hasequation 4x + 3y = 8.

A: Given information: The xy-plane has equation 4x + 3y = 8. Calculation: The equation of the plane is…

Q: Find the canonical or standard equation of the sphere, if it exists. 2x² +2y² + 2z² – 4x+12y–16z+6=0…

A: To write the canonical or standard equation of the below sphere. 2x2+2y2+2z2−4x+12y−16z+6=0

Q: Reduce the equation    4x^2 + y^2 + 4z^2 - 4y − 24z + 36 = 0    to one of the canonical forms of…

A: Reduce the equation    4x^2 + y^2 + 4z^2 - 4y − 24z + 36 = 0    to one of the canonical forms of the…

Q: Find 3 non-collinear points in the plane ax+by+cz+d=0. (Hint: Set two variables to zero for each…

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: Find an equation of the plane that passes through the point (1,2,3) and which is orthogonal to the…

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Q: Use the Lagrange multiplier method to find the point of the plane 4x-5y+3z=1 with less distance to…

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Q: Find an equation of the plane through the points P(0, 1, 1), Q(1, 0, 1), and R(1, 1, 0).

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Q: . (a) Show that as (x, y)→ (0, 0) along any straight line y = mx, or along any parabola y = kx², the…

A: Given : f(x) = x³y/(2x^6+y²)  To find : To prove that limit exist along parabola

Q: Classify the quadric surface ² - y² - x² + 4x = 5. 2 य

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Q: Find the points on the hyperboloid 9x^2  - 45y^2 + 5z^2 = 45 where the tangent plane is parallel to…

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Q: Find an equation for the plane in R® spanned by the points (1,5, 1) and (1, 4, 2).

A: We have to find the plane equation for two given points

Q: What is the center of X x² + y² – 30x + 221 = 0? Show all work for full credit.

A: Topic = Geometry

Q: Verify that the line r(t) = [1 + t, 1 - t, t] is part of the one sheeted hyperboloid x^2 + y^2 -…

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Q: Suppose a surface in three-dimensional space has the equation ax^2+bx+cy^2+dy=z  and it contains the…

A: A given set of points can be assumed to be on a surface given by an equation when the each of…

Q: Question 10

A: We have to find equation of plane that is parallel to the plane 6x-5y+8z+18 =0 and  the pale passes…

Q: Give an equation of the plane whose trace in the yz-plane has equation y = 6z, and in the xy-plane…

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Q: Find the point on the plane x+y+z%3D24 that is closest to the point (1,1,1).

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Q: 10. Determine the scalar equation of a plane that is perpendicular to the line: |1,2, 3| + t|-2,4,…

A: A normal of a plane perpendicular to the line will be parallel to the line. So the plane is:…

Q: The equation of a surface, with respect to an orthonormal coordinate system for three-dimensional…

A: Every surface equation with respect to an orthonormal coordinate system is written in a matrix form…