Questions on the use of algebraic techniques to prove geometric theorems.

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-1
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5answers
17 views

Find point on a line, given the line equation and distance from the origin [on hold]

Given the line $y=3x+6$, how to find the coordinates of the points on the line which are $9$ units from the origin?
1
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1answer
26 views

What's the parametric equation for the general form of an ellipse rotated by any amount? Thanks.

What's the parametric equation for the general form of an ellipse rotated by any amount? Preferably, as a computer scientist, how can this equation be derived from the three variables: coordinate of ...
0
votes
0answers
19 views

Finding an ellipsoid equation using its projected views

I want to find 3D equation of a falling droplet that I have considered it as an ellipsoid. I put two cameras, one in xy plane and another in zy plane to capture two projected views of the droplet and ...
-2
votes
2answers
17 views

Showing solution of normal line [on hold]

Find the equation of the line parallel to the line $ x-5y+5=0 $ and passing a distance $\pm 2$ from the origin.
1
vote
2answers
33 views

Distance of a Point from Hyperbola

Consider the part of hyperbola $H_{+}=\{(x,1/x)\colon x>0\}$ in the first quadrant, and $(a,b)$ any point in the plane (for sake of convenience, say $a,b>0$). If $(a,b)$ does not lie on the ...
1
vote
1answer
23 views

Locus of a point that satisfy a condition on the square of distances to two lines and their intersection

Find the locus of a point such that the square of its distance to the point of intersection of two perpendicular lines is equal to the sum of its distances to those lines. Assume $P(x,y)$ is any ...
0
votes
1answer
17 views

I want to know where I did wrong in finding the plane equation

I am asked to give 3 plane equation where the third plane will passes through the intersection of the first 2 planes and parallel to y axis. I came up with 2 plane equation which is also parallel to ...
2
votes
1answer
37 views

Geometric locus

The problem is: Let $A$, $B$ and $C$ be fixed points, and $α,β,γ$ and $κ$ are given constants, then the locus of a point $P$ that satisfies the equation $$α(AP)^2+β(BP)^2+γ(CP)^2=\kappa,$$ is a ...
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votes
2answers
15 views

Describing vector equation geometrically

How would I describe geometrically the vector equation: $$\vec{x} = s(0,2,1) + t(1,1,-1) ,\qquad s,t \in \Bbb R$$
2
votes
2answers
26 views

find equations of an ellipsoid axes

I have an ellipsoid with the center point at the Origin and the following equation: $$\alpha_1 x^2+\alpha_2 y^2+\alpha_3 z^2+2\beta_1 zy+2\beta_2 xz+2\beta_3 xy=1$$ How can I find the equations of ...
0
votes
0answers
16 views

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form?

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form such that when you draw 4 equilateral triangles the foot of the perpendicular of the equilateral ...
0
votes
1answer
17 views

Plotting Particular Conic Section

How would I plot $-2x^2 -2y^2 = 1$ on the x-y plane ? I believe it is an ellipse, since the coefficients have the same sign, I just don't know what the major and minor axes would be nor how to plot.
1
vote
1answer
27 views

General solution for intersection of line and circle

If the equation for a circle is $|c-x|^2 = r^2$ and the equation for the line is $n \cdot x=d $, and assuming that the circle and line intersect in two points, how can I find these points? Also as ...
0
votes
0answers
13 views

How to insert a simplifier assumption in our equations set to find an ellipsoid equation

Regarding the below question: Finding equation of an ellipsoid two projected views (two ellipses) is not enough to solve the equation set and find a unique ellipsoid. For example, I chose a ...
0
votes
0answers
28 views

Co-Ordinate Geometry Doubts

Facing trouble to find a solution for my problem. I am asking here. I hope I will get at least a clue for my question. I have two points $(x_1,y_1),(x_2,y_2)$ and the distance between them is $r$. The ...
0
votes
2answers
30 views

What is the d in the formula of a plane in $ R^3$

In algebra the formula for a line is $y=ax+b$ the $b$ moves the position of the line up and down the y axis. The formula for a plane is given to me as $ax+by+cz+d=0$ the $d$ must move the position of ...
1
vote
1answer
37 views

Using substitution to determine if a given point is on the line

Is it necessary to rearrange the equation of a line so that it is in the $y=mx+b$ form before using substitution to check whether a point is on the line? If yes, why? If no, why?
0
votes
2answers
24 views

Give the equation of a plane that crosses the axes at points equidistant from the origin

Give the equation of a plane that crosses the axes at points equidistant from the origin. I really have no idea how to start answering, since there are no points given to work with. Please help ...
1
vote
1answer
67 views

Finding equation of tangent of a circle that intersects the origin?

Given: circle with equation $(x-2)^2+(y-1)^2=4$. How to find equation of tangent line to the circle that intersects the origin? I easily found out that one of the tangents is $x=0$.
1
vote
1answer
56 views

Determine y-coordinate of a 3rd point from 2 given points and an x-coordinate.

I'm working through the "Calculus 1" Coursera course (offline version, so no forums) and am struggling with the following question in the topic on Limits: Consider points A=(-10,-4) and C=(8,5). ...
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votes
2answers
52 views

Find the other 2 points of a rectangle? [closed]

$PQRS$ is a rectangle with vertices $P(-4,-1)$ and $Q(-6,5),$ and $PQ=2(QR).$ Find the coordinates of $R$ and $S$? I'm so stuck please help! There are 2 answers for each point. almagest has the right ...
2
votes
1answer
46 views

A $k+1$-sphere containing a $k$-sphere and a point.

Earlier I asked a question on whether it is possible to find a sphere passing through a circle and a point non-coplanar to it. I wanted to know whether this was possible to do in higher dimensions. ...
0
votes
1answer
30 views

Algebraic proof for sphere/circle overlap formula

Two spheres or circles denoted by center position vector and radius $ p_0, r_0$ and $p1, r_1$ will overlap if $$ |p_0-p_1| < r_0+r_1$$ I understand geometrically why it works, but how would one ...
3
votes
3answers
60 views

45 degree rotation of the line $y=-3x+1$?

Currently working on problems in a textbook for Senior Maths (Year 11 Maths C, named 'Maths Quest - Maths C for Queensland), however I'm currently at a problem where my answer, despite attempting it ...
0
votes
1answer
35 views

Find the three dimensional line that goes through point p and is perpendicular to a plane

I am given the point $P(1,0,6)$ and I need to find a line that goes through $P$ and is also perpendicular to $x+3y+z=5$. Background info: I've gotten the help I needed now but when I started I was ...
0
votes
1answer
17 views

Find the locus of the the vertex A.

Consider $\triangle ABC$. BC lies on a line passing through $(g,f)$. The pair of straightlines $(x+y)(x-9y)=0$ are the perpendicular bisector of sides AB and AC of $\triangle ABC$. Find the locus ...
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votes
1answer
34 views

If $ 2y\cosθ=x\sinθ $ and $2x\secθ−ycosecθ=3$,then the relation between $x$ and $y$ is. [closed]

If $ 2y\cosθ=x\sinθ$ and $2x\secθ−y\operatorname{cosec}θ=3$,then the relation between $x$ and $y$ is. Note:$2x/\cosθ-y/\sinθ=3$ $2x\sin θ-y\cos θ/(\sin θ\cos θ)=3$ $4y\cos θ-y\cos θ=3\sin θ \cos θ$ ...
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votes
1answer
34 views

The locus of points with given sum of squares of distances to two fixed points

$A(a,b)$ and $B(b,-a)$ are two fixed points. If $P(x,y)$ is a moving point such that $$|AP|^2 + |PB|^2 = |AB|^2 \tag1$$ prove that $x^2 + y^2 =(b-a)(x+y)$. So far I tried to use distance formula ...
0
votes
1answer
45 views

graphing hyperbola algebra problem

I have the hyperbola from a textbook 9x^2 - 18x - 16y^2 - 64y = 91 It is supposed to become: ((x-1)^2) / 4 - ((y+2)^2) / (9/4) = 1 I cannot get this though, I arrive at: ((x-1)^2) / 4 - ((y+2)^2) / ...
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vote
2answers
42 views

Finding the intersection of an xy-plane in a 3D-Coordinate System

I found the equation of a sphere that has a center of $(1,-12,8)$ with a radius of 10 and I got the following equation: $(x-1)^2 + (y+12)^2 + (z-8)^2 = 100$ As for finding an intersection for the ...
0
votes
1answer
38 views

Area of a triangle - straight lines

Question: What is the area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight lines $x^2 - y^2 + 2y = 1$. Well I really have no idea how to even start the ...
5
votes
1answer
62 views

Shortest path between two points via two disks

Hallo everybody, I have the following problem regarding shortest paths in $R^2$. Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture. I am looking for a path from ...
0
votes
1answer
37 views

Show that $f(x)$ satisfy the differential equation

Given a curve $C=\{(x,f(x)\in \mathbb{R}\times\mathbb{R}\mid x\in(r_1,r_2)\}$ with has the following property.(f(x) is $C^3$-function) At any point $(a,f(a))\in C$ if we change coordinate system by ...
0
votes
0answers
9 views

Extract equations of dependency between two projected views

Regarding question Finding equation of an ellipsoid, the answer says that we have the following equation between projections on XY & XZ plane: $$\frac{Z_3^2}{Z_2} - Z_1 = \frac{Y_2^2}{Y_3} - Y_1$$ ...
1
vote
0answers
38 views

How to solve the sets of equations to find the matrix of coefficients for an ellipsoid

Regarding the below question: Finding equation of an ellipsoid I have two more questions: 1- In the "update" section of answer provided by @achillehui I can not understand the method he described for ...
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votes
0answers
42 views

How to show that if a complex function is analytic then it is infinitely many times differentiable geometrically? [duplicate]

I am going through the theorem which proves that if a complex valued function is analytic than it is infinitely many times differentiable. But I am not sure how to explain this geometrically without ...
2
votes
3answers
64 views

Finding the equation of a line whose segment is intercepted between axes

The question is: Find the equation of a line through (-2, 5) and whose segment intercepted between axes in the 2nd quadrant is 7√2 I have two graphs in mind but I don't know which one is correct. The ...
0
votes
2answers
21 views

proves of parametric curves via parametric equations

Hi could anyone help me with this problem. An astroid is given by the equation $$x^{2/3} + y^{2/3} = 1.$$ Prove via parametric equations that the length of a piece of a tangent line between the ...
0
votes
0answers
28 views

Holomorphic and meromorphic functions on Riemann surfaces

On any domain $\Omega\subset \mathbb{C}$, the set of all holomorphic functions form an integral domain. Its field of quotient is the set of all meromorphic functions on $\Omega$. However this is not ...
1
vote
1answer
16 views

Vectors in 3 dimensions

If $a$ is a vector that makes equal angles with ${\mathbf i},{\mathbf j},{\mathbf k}$ and has magnitude $3$, then find the angle of $a$ with either of these unit vectors? Wouldn't the answer simply ...
5
votes
2answers
50 views

How is Cartesian coordinate system related to his philosophy

In 1637, Rene Descartes published his famous monograph about philosophy "Discourse on the Method of reasoning well and Seeking Truth in the Sciences", and analytic method of geometry has been come up ...
1
vote
2answers
170 views

Finding equation of an ellipsoid

Consider I have an ellipsoid (let say an egg) lies in a general form in 3D space. Suppose, I have the equations of two projected views of this egg (e.g. one projected view on x-y plane and another one ...
1
vote
1answer
31 views

How to find the angle between two vectors?

Here, I would like to describe my requirements .. Let's say we have two vectors named $\bf A$ and $\bf B$. Two vectors are in different magnitude and opposite directions and lay on different planes. ...
0
votes
2answers
38 views

How to show that a given line has a certain equation?

Say line $A(3,0)$ and $B(0,2)$ How do I 'show' that they have equation $2x + 3y - 6 = 0$?
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vote
0answers
42 views

Questions about circle

I found the following problem from a book. Let A = (-1, 0), B = (1, 0) and k = a constant which is not equal to 1. C(x, y) is a variable point such that AC = kBC. Find the locus of C. The ...
0
votes
1answer
79 views

How is the curve with equation $1/x^4 + 1/y^4 = 1$ called?

Well what is the graph for $$\frac 1{x^4} + \frac 1{y^4} = 1$$ called? According to $ Wolfram-Alpha$: http://www.wolframalpha.com/input/?i=plot+1%2Fx%5E4%2B1%2Fy%5E4%3D1+and+y%3Dx+and+y%3D-x ( ...
2
votes
3answers
49 views

Pair of straight lines

Question: Find the equation of the bisector of the obtuse angle between the lines $x - 2y + 4 = 0$ and $4x - 3y + 2 = 0$. I don't even know how to proceed here. I know how to find the angle between ...
3
votes
1answer
17 views

If $P=(x_0,y_0)$ is a point in a focal chord of the parabola $x^2=4py$ then find the coordinates of the other point

$\textbf{Exercise:}$ If $\overline{PQ}$ is a focal chord of the parabola $x^2=4py$ and the coordinates of $P$ are $(x_0,y_0)$, show that the coordinates of $Q$ are $$ ...
1
vote
1answer
41 views

Show an equation of a line passing through $P$ and parallel to the line given by $ax+by+c=0$.

Question: A person considers lines on the plane $\mathbb{R^2}$ to be solutions of equations of the form $ax+by+c=0$, where $a,$ $b,$ and $c$ are fixed reals satisfying $a^2+b^2\neq0$. Give a point ...
1
vote
2answers
25 views

How to define a cloud of points relative to a vector path?

I've been researching and playing with examples of particle clouds in a graphics visualization. Most use shape geometries to define a field of particles, or parameters for distributing them randomly ...