Find the equations of planes that just touch the sphere (x−5)^2+(y−3)^2+(z−1)^2=9 and are parallel to (a) The xy-plane: _____ and _____   (b) The yz-plane: _____ and _____     (c) The xz-plane: _____ and _____

Q: Write the equation of a sphere that has a center at C(2, 5,-3) and contains points D(4, 1, 1) on the…

A: Given Data: The centre of the sphere is: C(h, j, l)=C(2, 5, -3) The point at the edge of the sphere…

Q: A city is described parametrically by the equation r = (ai + bj+ck) + svi + tv₂ where v₁ = 3i+j+ k…

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Q: Find an equation for the plane containing the three points (0, 1, -1), (1,0,1), and (2, 1, 0). Where…

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Q: Find an equation of the plane that crosses the x-axis at 3, the y-axis at 2, and the z-axis at 5.

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Q: Find the equation for the plane through the points Po(5, - 4, - 4), Qo(2,0. – 2), and Ro(4,1, – 2).…

A: Introduction: A plane is a three dimensional geometrical structure. When there is a projection of a…

Q: Consider the sphere (x − 4)² + (y − 3)² + (z − 5)² = 25 (a) Does the sphere intersect each of the…

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Q: Find an equation of the plane through the origin that is parallel to the plane 4x-3y+5z+13=0

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Q: You are given the following points: A=(−4,20,16)A=(−4,20,16), B=(20,0,7)B=(20,0,7),…

A: Given- (x,y,z) coordinates of three points :A =(-4,20,16) B=(20,0,7) C=(8,-15,-19) to find- closest…

Q: dw. Findequatian fthe plan that Passer thraugh the originand thePaint P and parallet te the tineof…

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Q: A laser pointer is situated at the point (3, 1, 4) and is pointing a linear beam in the direction…

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Q: (1) Find an equation of a plane that goes through (-1,2,6) and is perpendicular to (2, 1, –3).

A: It is given that the perpendicular vector to the plane is given as,  and plane passing through,…

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.