I need help answering the following question: Suppose we have a collection of points in the Cartesian​ plane, and we are told that ​ f(x)  is a function which passes through all of them. Must it be true that the​ x-coordinates of the points are all​ different?  Why or why​ not? Must it be true that the​ y-coordinates of the points are all​ different?  Why or why​ not?

Q: 2. An is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two…

A: We know that, The sum of distances to any point on the ellipse x,y from the two foci c,0 and -c,0 is…

Q: Of all the points on the graph of z = 10 - x^2 - y^2 that lie above the plane x + 2y + 3z = 0, what…

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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: Use the sub no spaces: T-2y = - %3D 3x + 2y

A: Given that: x-2y=-10           ........(1)3x+2y=-14      ..............(2)

Q: 5. Prove that whole length of the curve x3 + y2/3 = a/3 is 6a.

A: Given that x23+y23=a23 To find the length of the curve:…

Q: Find a plane containing the point (7,-8,8) and the line F(t) = +t

A: Given that, the point is 7,-8,8 and the line is r→t=-6,-8,-7+t6,8,4. From the given line, the point…

Q: +3そ20 22 Prove that the line X21+2t ニー+3t is perpendicular to the plane X-2y+zz6

A: As per our guideline we are allowed to do 1 question at a time. Please post other questions next…

Q: (5xy --- 6x(2^

A: Given,        F=(5xy-6x2)i+(2y-4x)j and   y=x3 from the point (1, 1) to (2, 8).

Q: Use the Cauchy-Riemann PDEs to determine all points z = x + iy ∈ C in which the following functions…

A: First separate real and imaginary components. Then apply C-R equation.

Q: Prove that the line x +1 = 글(-6) - lies in the plane 3x + y – 2 = 3.

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Q: 2. Prove that the following curves are intersecting with a right angle: 5y-2x+ y –x*y =0 2y+5x+x*…

A: Given curves are  5y-2x+y2-x2y=0and2y+5x+x4-x3y2=0

Q: 3. Describe the level surfaces of the function. f(x, y, 2) = x² + 4y? + 42?

A: We Know that,  A level surface of a function w = f(x, y, z) with three variables is a surface f(x,…

Q: If all of the level curves of a function f(x, y) are parallel lines, then the function must be…

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Q: Find the distance between the planes 5x - 3y + z = 9 and 5x - 3y + z = -6.

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Q: Find the distance between these two planes x + 2y –' 2z = 3 and 2x + 4y - 4z = 7.

A: Devide the equation of 2nd plane by 2 .we get that both planes are parallel so just apply formula…

Q: If f(x,y,z) = x2 + y2, what is the locus of points in space for which grad f is parallel to the…

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Q: 22 Prove that the line Z=2+46 is perpendicular to thme plane X-2y+2=6

A: As per our guideline we are supposed to do 1 question at a time. Please post other questions next…

Q: Prove that the following curves have infinitely many rational points: y? = a3 – 3x? + 3x – 1, y? =…

A: The given curve is y2=x3-3x2+3x-1. Rewrite this as follows. y2=x3-3x2+3x-1y2=x-13y=x-13         or…

Q: Q5: Consider the following subsets of real plane R²¹ Determine with graph: 1-Arewise connected A=…

A: Here we have a common point in both of the sets. Also thos two sets are individually connected.

Q: Find the point at which the line f(r) = -z+1intersects the line g(z) = 4x +1 %3D

A: At point of intersection both the functions will have the same value therefore f = g

Q: Find the points on the graph of z = xy + 8y where the tangent plane is parallel to 2x + 5y + 2z = 0.…

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Q: Find the image of the line y =-x under the mapping = f(z) =2² + 4z %3D %3D

A: We need to find the image of the line y = - x under the given mapping.

Q: 1. Sketch the graph of the domain of the function f (x, y)= sq x / x−y​ 2. Compute the equation of…

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Q: Prove that the following curves are intersecting with a right angle: 5y-2x+ y -x²y =0 2y+5x+ x* –…

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Q: Prove that the curves o_1={|z| =2} and o_2= {|z-i|=5} are homotopic.

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Q: OShow that every point (b, c) in the plane lies on a tangent line to the graph of y = x³.

A: Consider the given information. y=x3 Find the derivative of the function. dydx=ddxx3y'=3x2

Q: Verify Green's Theorem for: f4y³dx-2x²dy where C, oriented anti-clockwise, is composed of the…

A: To find the integral ∮C4y3dx-2x2dy where C is the positively oriented closed curve x2+y2=9, y≥0 and…

Q: | Let w=z² Find the image of the first quadrant in Z-Plane 2

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Q: Among all the points on the graph of z = 10-x2 -y² that lie above the plane r+2y+3z = 0, ind the…

A: Given that z=10-x2-y2 and x+2y+3z=0 Find the point which is farthest from the plane.

Q: 9 find the cqu. f Plane that Contains the two line Ś: X-1=3+ 6=そ X=リニで-4 and Passes through origin?…

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Q: 9. Suppose that a : R³ → R³ by perpendicular projection onto the plane 3x – 2y + z = 0. Find ↑

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Q: 4) Suppose that the level curves of a function z = f (x, y) consists of straight lines. Must the…

A: Given that level curve of a function z=f(x,y) consists of straight lines.

Q: Show that for every real number k the plane x - 2y + z + 3 + k(2x - y - z + 1) = 0 contains the line…

A: Show that for every real number k the plane x - 2y + z + 3 + k (2x - y - z + 1) = 0 contains the…

Q: 3. Discuss how to evaluate f () dz , where C is a simple closed curve C=10 enclosing the origin.

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Q: 4. Prove that the following curves are intersecting with a right angle: 5y-2x+y²-x²y = 0 2y +…

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Q: Choose a set S of four distinct points in R3 such that aff S is the plane 2x1 + x2 – 3x3 12. Justify…

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Q: Compute the length of the curve: f (x) = x² 2x, 0<x<2

A: Given:- f(x) =x2 -2x  To find:-  Length of curve over interval [0, 2]

Q: Let F(x, y) = <3x?y3 + y", 3x°y² + y^ + 4xy³,. %3D Evaluate F. dr where C is the line segment from…

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Q: List the points in the xy-plane, if any, at which the function z = 7+ ((x-4)² + (y+3)²)´ is not…

A: The given function is as  z= 7+((x+4)^2 +(y+3)^2)^2  We have to check the that this function is…

Q: Prove that three points (x1, yl), (x2, y2) and (x3, y3) are collinear ( on the same straight line)…

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Q: Assume that f:R² → R² is a continuous bijection such that the image of each line is a line. Show…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find the set of points that is satisfied the condition described by |z – 2i|+ |z + 4i| = 10

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Q: 5. Show that the following orthogonal families of curves: are x² + y² : = ax, x² + y? = by

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Q: A Show that the curve F(t) = lies in a plane.

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Q: In the following problem, what is f (the function to optimize) and what is g (the restriction)?…

A: The surface equation is xy=3x2-7z and the point is 1, 3, -2. Consider the point on the surface as x,…

Q: Find the points on the graph of z = xy + 8y-1 where the tangent plane is parallel to 6x + 5y + 6z =…

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Q: A student is graphing the function y= Vz+4 on the coordinatė plane. Which statement will help the…

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Q: Describe geometrically the set of points z satisfying the following condition:|z−2i|=3 .

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.