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

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2
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2answers
295 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
89 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
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1answer
21 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 ...
3
votes
1answer
54 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 ...
0
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2answers
29 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
150 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 ...
3
votes
1answer
477 views

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle. Also find the center and radius of the circle. Here's my solution, I'm ...
2
votes
1answer
70 views

Find the circle which passes through two points

Find the equation of a circle which passes through $(4,-3)$ and $(-3,-4)$ with radius $5$. I tried putting the $x$ and $y$ into the equation $(x-h)^2 + (y-k)^2 = r^2$, but then I don't know how to ...
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0answers
38 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
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1answer
18 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.
2
votes
1answer
364 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 ...
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2answers
38 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 ...
2
votes
1answer
56 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?
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vote
1answer
218 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 $x=0$.
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vote
1answer
2k 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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2answers
174 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
59 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
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1answer
84 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
1answer
55 views

How to compute coordinates of a point that intersects an sphere

Hi all. Is there a way to compute the S(x,y,z), given the following information: A(x,y,z) e = elevation (from the line AS) Az = azimuth (over A). Perpendicular to x axis. Can vary from 0 tp 360. ...
3
votes
3answers
113 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
450 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
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1answer
102 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
104 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 in ...
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1answer
78 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
1k 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 xy-...
0
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1answer
285 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 ...
6
votes
1answer
91 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 $p$ ...
0
votes
1answer
45 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 ...
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0answers
56 views

How to solve a sets of equations

I capture each of the projected views of a droplet through a high speed camera (one in xy plane and one in zy) and then analyze the frames by image processing to find the related equations for each ...
2
votes
3answers
909 views

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

The question is: Find the equation of a line passing through $(-2, 5)$ and whose segment intercepted between axes in the 2nd quadrant is $7\sqrt{2}$ I have two graphs in mind but I don't know which ...
0
votes
2answers
34 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 ...
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vote
1answer
19 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 ...
9
votes
3answers
4k views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
5
votes
2answers
110 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 ...
2
votes
1answer
338 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
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1answer
136 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. ...
2
votes
3answers
1k views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
0
votes
2answers
85 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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votes
2answers
113 views

“Conic sections” that are really just two straight lines

My teacher was teaching co-ordinate geometry and today he said that the following equation will always represent a conic section:$$ax^2+by^2+2hxy+2gx+2fy+c=0$$ Then he said that if the determinant of ...
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0answers
59 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
90 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
89 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
120 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 $$ \left(\frac{-4p^2}{x_0},\frac{p^...
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vote
1answer
162 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 $P=...
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vote
2answers
37 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 ...
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0answers
44 views

Equation of a line through a point and another line

I need to get the equation of a line that passes through the point Q(6, 3, 2) and intersects: $$L: (1, -1, 4) + t(0, -1, 1)$$ and forms an angle of 60° What I did so far: The direction vector of L ...
3
votes
2answers
48 views

Parallel plane that contain lines

I have the lines: $$L_1: \frac{x-3}{2}= \frac{y+5}{-3} = \frac{z+1}{5} ,$$ $$L_2: \frac{x+1}{-4}=\frac{y-1}{3}=\frac{z-3}{-1}$$ I need the equations of the parallel planes $P_1$ and $P_2$ that ...
0
votes
1answer
263 views

Finding Shortest distance between a Sphere and Ellipsoid?

Suppose that ,I have a Sphere and an ellipsoid as Sphere: $(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$ Ellipsoid: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
4
votes
3answers
261 views

What is the cone of the conic section?

Given the general (real valued) equation of a conic section: $$ A x^2 + B xy + C y^2 + D x + E y + F = 0 $$ Then what is the circular cone associated with it ? Is it unique ? And is there a way to ...
2
votes
2answers
64 views

Analytic Geometry: One sheeted hyperboloid

Good afternoon! I have a question about analytic geometry. I don't actually know if the answer is quite simple, and I missed something while revising, or if it is actually more complicated than I ...