What kind of conic section is given by the quadratic form ? -11x,? + 84x,x2 + 24x2? = 156 Transform it to principle axis. Express x" = [x1 x2] in terms of the new coordinate vector y" ly, y2]

Q: What is the parametric equation of line through the point (1,2, -1) and perpendicular to the plane…

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Q: Find the distance from the point (3, 2, 5) to the plane whose general equation is 2x + 3y - z = 0.

A: we have to find the distance from the point (3,2,5) to the plane whose general equation is 2x+3y-z=0…

Q: Determine the orthogonal trajectories of the hyperbola 3xy= 5 at the point (2, 5/6). Sketch the…

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Q: 1. Find all points Po on the ellipsoid 2x2 + 3y? + 2z2 plane x – 3y + 2z = 5. = 22 at which the…

A: To find points P0 on the ellipsoid 2x2+3y2+2z2=22   The tangent plane is parallel to the plane…

Q: Find a parametrization for the line segment beginning at P,(4, - 7,- 7) and ending at P2(0, - 7,2).

A: The line segment beginning at P1<4, -7, -7> and ending at P2<0, -7, 2>

Q: A point moves along the intersection of the elliptic paraboloid z = x² + 3y² and the plane y = 1. At…

A: Point moves along the intersection of the elliptic paraboloid z=x2+3y2 and the plane y = 1. We have…

Q: Find the distance from the point Q(1, 2, -4) to the plane 2x - 5z = 5.

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Q: Show that the cylindrical equationr 2(1 − 2 sin2 θ ) + z 2= 1is a hyperboloid of one sheet.

A: The cylindrical co-ordinates is converted into rectangular co-ordinate as x=rcosθ         ⋯⋯1y=rsinθ…

Q: What kind of conic section is given by the quadratic form ? -11x,2 + 84x,x2 + 24x,2 = 156 Transform…

A: Given Quadratic Equation is, -11x12+84x1x2+24x22=156 Matrix Representation form is, A=-11424224 Find…

Q: Determine the parametric equations of the curve of of the ellipsoid 4x² + y2 + 4z2 = 4 and the…

A: Solution: Given equation:ellipsoid: 4x2+y2+4z2=4; and half−cone: y2=x2+2z2 To find: parametric…

Q: Give a scalar equation for a plane at distance 9 from the plane P with equation -2x-2y+z = 4.

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Q: Show that the parabolas $r=c /(1+\cos \theta)$ and $r=d /(1-\cos \theta)$ intersect at right angles.

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Q: Give a vector parametric equation for the line that passes through the point (3, 5, –5), parallel to…

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Q: Find the equation of the tangent plane to the paraboloid z = 2x2 +y? at the point (1,2,6). Select…

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Q: Find all points on the ellipsoid 2x² + 3y² + 4z² = 9at which the plane tangent to the ellipsoid is…

A: Explanation: Given that,  Ellipsoid:2x2+3y2+4z2=9Plane:x-2y+3z=5⇒15x-25y+35z=1   Suppose point of…

Q: Where does the parametric line ( 2 + 2 t , t , t ) intersect the plane x+y-3z=4?

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Q: Q1/ plot x2-y2=z (Elliptic paraboloid) in 3D.

A: Topic:- plot in 3-D

Q: Show that the cylindrical equation ,2(1 – 2 sin² 0) + z² = 1 is a hyperboloid of one sheet.

A: Given a cylindrical equation r21-2sin2θ+z2=1 To show that this equation is a hyperboloid of one…

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: The parametric equation of the segment [AB] with A(1,0,5) and B(3,1,2) is: O x=1+2t y%3Dt z-5-3t -o<…

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Q: y -6- 4- 3- 2- 1- -3 2 -1 0 12 3 4 5 6 -1- -2- R -3- your work on question 10, classify this…

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Q: Find an equation for the plane through the point (3, 4, −4) and parallel to the plane defined by 4x…

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Q: 3. Find an equation of the tangent plane to the paraboloid defined by z= 5x +4y? at the point…

A: The given equation is; z=5x2+4y2 at the point (-1,1,9).

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

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Q: Derive the orthogonal trajectories of the family of hyperbolas ??=?.

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Q: Find the point P nearest to Q = (−1, 3, −1) on the planex − 4z = 2

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Q: Find the distance between the point (0, 0, 0) and the plane 2x + 3y + z = 12.

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Q: Find a parametrization for the line through the point (a, b) having slope m.

A: Given that slope = m , this can be written as &lt;1,m&gt; vector. And point= (a,b)

Q: A point moves along the intersection of the elliptic paraboloid z = 9x + 7y and the planex = 1. At…

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Q: Find the equation of the plane containing the point (4, -1, 3) that is parallel to the plane 2x - 3y…

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Q: Give a parametric representation for the cylinder x^2+z^2=121

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Q: 4. Find the equation of the tangent plane at the point (-2, 1, -3) to the ellipsoid 22 + y? + = 3 4

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Q: 3. Find the area of the surface cut from the nose of the paraboloid x = 1-y²-z2 by the yz-plane. Do…

A: Surface area in yz-plane

Q: Find a parametrization for the line SEGMENT between (- 5, 7, 6) and (2, – 6, – 9). |

A: Parameterizing a line segment, which joins the points (x1,y1,z1) and (x2,y2,z2), helps in obtaining…

Q: Find the distance between the point (3, −2, 4) and the plane 2x − 5y + z = 10.

A: Given: The equation of a plane is 2x-5y+z=10. To determine: The distance between the point 3,-2,4…

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

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Q: find The unit Vectors tanent and normal to The ellipse x+2y= 6 at The Point (2,1).

A: Given, ellipse x^2+2y^2=6   point=(2,1)

Q: Which of the following is the equation with center (-9, 8) Major axis = 10? %3D 20 and minor axis…

A: To find the equation of the vertical ellipse with center (-9,8) . Major axis = 20 and minor axis =…

Q: Give a vector parametric equation for the line through the point (−1,−1) that is perpendicular to…

A: Vector equation of line which passes through point r0 and perpendicular to line a+bt,c+dt is given…

Q: Give a scalar equation for a plane at distance 9 from the plane P with equation -2x+y-2z = 3.

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Q: Find an equation for the tangent plane to the hyperboloid x2+y2−z2=1 at the point (3,1,3).

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Q: Find a parametrization for the ray (half line) with initial point (-4,3) that passes through the…

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Q: 1. Determine the orthogonal trajectories of the hyperbola 3xy = 5 at point (2,5/6). Sketch es. %3D

A: Topic = Geometry

Q: Consider the conic below x² + 4xy+ 4y² – 2/5x + /5y = 0 and determine what is asked. Translation…

A: Here, the given equation is x2+4xy+4y2-25x+5y=0. Firstly, let's evaluate the angle of rotation, θ.…

Q: - Show that the distance r from the origin O to the point P(Coshu,Sinhu) - on the hyperbola x-y 1 is…

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Q: Flnd the parametric equation of the line of ntersection of the planes and

A: First find normal vector then take cross product and use formula

Q: What solid of rotation is created here if the axis of rotation is the x-axis? -2

A: to find:- type of solid generated after rotation about the x-axis.

Q: Find the general equation of the plane through the point (3, 2, 5) that is parallel to the plane…

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Q: Find a point on the ellipsoid 3x2 + y2 + 3z2 = 1 where the tangent line is parallel to the plane…

A: Given: The equation of ellipsoid is 3x2+y2+3z2=1 and the equation of the plane is -12x+2y+6z=0.…

Q: perpendicular

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