8. The line given parametrically by r(t) = (1+2t, 2+5t, 1– 3t) for t E R passes through a plane at the point (1, 2, 1) and this line is perpendicular to the plane. Find the equation of the plane. Write your final answer in the form ar + by + cz = d.

Q: 20. Find the point at which the line a = 1+ 2t, y = 4t, z = 2 –t intersects the plane r+y – 2z +1=…

A: Given that, equation of line:- x=2t, y=4t, z=2-t Equation of Plane:- x+y-2z+1=0

Q: The equation of the plane that contains the point (1, 2, 3) and the line x = 4t, y = t, z = t is…

A:

Q: Find the equation for the plane through Po(-6,3,7) perpendicular to the following line. x= -6+t, y…

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Q: Which of the following is the equation of the plane that is tangent to the surface 2 x² + y? – x z –…

A: Consider the provided information, The provided surface, 2x2+y2-xz-3z=1 Find the partial derivative…

Q: (b) Find an equation of the plane that contains the line x = 1 + t, y = 2-t, z = 4 - 3t and is…

A:

Q: 2.) Find the Symmetric form of the equation of the line of in tersection of the two Planes X-Y +Z…

A: Let a, b and c be the direction ratios of the required line. Since the required line lies in both…

Q: 2. Give the symmetric equations of the line l perpendicular to the plane T : 5x– 2y = z+4 and…

A: we have to find equation of line.

Q: Given S as a quadric with equation 4x^2 - (y-3)^2 + 4z^2 = 4 Find a generating curve for S (as a…

A: Given S as a quadric with equation 4x^2 - (y-3)^2 + 4z^2 = 4 Find a generating curve for S (as a…

Q: The equation of the plane passing through the point (0, 0, 4) and perpendicular to 4 is: the line 4…

A: Equation of plane

Q: 33 and Complete the parametric equations for the line where the planes – 3x – 9y – 12z = - 11x + 2y…

A:

Q: Find an equation of the plane tangent to the following surface at the given point. z=6-3x- 3y; (2,3,…

A:

Q: Give the symmetric equations of the line l perpendicular to the plane T : 5x – 2y = z+4 and passing…

A:

Q: 3. Sketch the points determined by the following. (a) Iz+1-i = 2 %3D (b) Im(2z +i) +1=0

A:

Q: 3. Find the general form of the equation of the plane containing the point Q(0, -2, 10) and the line…

A:

Q: 8. Find a parametric representation of the line through the point P(1,0,4) and perpendicular to the…

A:

Q: H.W: Find the distance from S (1,1,3) to the plane 3x+ 2y+ 6z = 6

A:

Q: Find an equation of the plane tangent to the following surface at the given point. z= 6- 3x - 2y;…

A: The equation of tangent plane to the surface z=fx,y at the point a,b,fb can be calculated using the…

Q: 8. The position of a particle moving in the xy-plane is given by the parametric equations x =…

A:

Q: Find the equation of a plane through the point (1, – 1, 6) and orthogonal to the line x(t) = 7 + 4t…

A: Required plane is orthogonal to the given line, say L: x(t)=7+4ty(t)=-3-4tz(t)=-1-6t also, the plane…

Q: Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2t,…

A: Explanation: Given that,  Line 1: x=1+ty=1+6tz=2t Vector 1: 1,6,2 Line 2: x=1+2sy=5+15sz=-2+6s…

Q: (b) Calculate the distance from the point P(3,–1, 2) to the plane with equation 2x – y +z = 4.

A:

Q: Find an equation of the plane containing the point (1,2,-2), and the line: x= 2t, y= 3-t, and z…

A: To find the equation of plane we need to find its normal vector. To find the normal vector, we…

Q: 2. Find an equation of the plane that contains the line r = 1 +t, y = 2 - t, z = 4 – 3t, and is…

A:

Q: 1. Find the parameter λ in the parametric form of the line l z+2 6:2-3-9-1-2-20 l: = X y so that…

A:

Q: s) Find the parametric equations of the line that is perpendicular to the plane 2x +y+3z = 4 through…

A:

Q: 9. Find the equation of the plane through P(1,1,4) that is parallel to the plane x+ 5y – z + 8 = 0

A: The scalar equation of parallel planes differ only by a constant Therefore, equation of the required…

Q: 6) Consider the plane P : x+y – z = 0 in K³ that contains both the points P1(1,0, 1) and P2(-1,1,0).…

A: Given : plane P : x + y - z = 0  & Points P1(1,0,1) & P2(-1,1,0) To find :  1. Equation of…

Q: Find the intersection of the line r(t) = 3t + 2, 1, −7t and the plane 2x − 3y + z = 5.

A: r(t)=3t+2, 1, -7t and the plane 2x-3y+z=5

Q: 2. Consider the line r(t) = (4t+1,3, 2 - t) and the ellipsoid + + z? = 1. What is the distance from…

A: Consider the line  and the ellipsoid To find  distance from the point (-3,3,3) on the line to the…

Q: Find the equation of the plane tangent to the following surface at the given points. 2 x +y -z - 52…

A: See the details solution in below

Q: Find an equation of the plane tangent to the following surface at the given point. z= 3- 3x -y:…

A:

Q: 3. Find the equation of the plane containing both the point (1, 2, 3) and the line r(t) = (t, 2t+1,…

A:

Q: 14. Find the trace of the plane 6x 3y + z = 4 in the xz-plane. 15. Determine whether the curve given…

A:

Q: 2) Find the equation of the plane that is orthogonal to the planes 3x-2y-2z=1 and x-3y-10z = 4 and…

A: Given: The equation of two planes 3x-2y-2z=1 and x-3y-10z=4 and the equation of one line…

Q: 6) = Find the point at which the line intersects the given plane. x =3 - t, y = 2 + t, z = 5t; x - y…

A: Equation of a Line in Parametric Form: where are functions of and is the parameter The equation of…

Q: Find the equation of the plane that passes through the point (2, 1,2) and is tangent to the surface…

A:

Q: (b) Find the points on the line r(t) = (t, 1+2t, 1-2t) whose distance from the plane 2x - 2y+z=1 is…

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Q: Given the following two lines: D:: (x,y,z) = (2,0, 0) + k (0,3,0) and Dz: (x,y,2) = (2,0,2) +k…

A:

Q: ) Find the equation of the plane passes through the point P(2, 2, 3) and perpendicular to the two…

A: To find the equation of the plane that passes through the point P (2, 2, 3) and perpendicular to the…

Q: Find the equation of the plane that passes through the point (2, 1, 1) and is tangent to the surface…

A: We have to find the equation of the tangent plane.

Q: 13. Consider the point P(3,2,1) and the plane given by the equation 3x +2y+z=0 on the plane closest…

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Q: Find the plane determined L1: x=-1+t 12: x=1-4s by the intersecting lines: y=2+t y=1+2s 2=1-1-3-1-00…

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Q: Suppose a particle P in R' is traveling upward in a straight line which is perpendicular to the…

A: Given: Suppose a particle P in R3 is traveling upward in a straight line which is perpendicular to…

Q: Q2) Find the distance from the point P (1, 2, 3) to the plane x-3y + 2z= 4.

A:

Q: A curve in the plane is defined parametrically by the equations x = t +t and y t* + 2t2. An equation…

A: The equation of the tangent line is: y-y1=mx-x1 Where m is the slope of the tangent line. And The…

Q: nsider the equation of the surface S given by 2x. + 3XYP + y2 wher mber and ß is an impair number ,…

A: In this question, the concept of the derivative is applied to find out the tangent to the plane.…

Q: 4. Consider the equation 16y" - 4 + 2² = 16. By first deseribing all of its traces, sketch the…

A: Given,  The equation, 16y2 -4x2 + z2 = 16

Q: Find the point on the plane x – 3 y – 2 z = 14 that is closest to the point (0, 3, 3) .

A: We have the plane x-3y-2z=14 The normal vector to a plane ax+by+cz=d is a,b,c Hence, the normal…

Q: 6. Find the projection of u = -7 onto the -2 line e of R given by the parametric equation l = tũ, -1…

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.