Skip to main content

Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.

Filter by
Sorted by
Tagged with
39 votes
17 answers
228k views

Get the equation of a circle when given 3 points

Get the equation of a circle through the points $(1,1), (2,4), (5,3) $. I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?
JohnPhteven's user avatar
  • 2,037
231 votes
23 answers
218k views

How to check if a point is inside a rectangle?

There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
Freewind's user avatar
  • 2,535
63 votes
12 answers
46k views

Is there an equation to describe regular polygons?

For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
Vincent Tan's user avatar
66 votes
6 answers
147k views

What is the general equation of the ellipse that is not in the origin and rotated by an angle?

I have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?
andikat dennis's user avatar
34 votes
2 answers
76k views

Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
Alekxos's user avatar
  • 543
31 votes
6 answers
49k 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 ...
najayaz's user avatar
  • 5,479
11 votes
4 answers
3k views

How do I prove this method of determining the sign for acute or obtuse angle bisector in the angle bisector formula works?

The formula for finding the angular bisectors of two lines $ax+by+c=0$ and $px+qy+r=0$ is: $$\frac{ax+by+c}{\sqrt{a^2+b^2}} = \pm\frac{px+qy+r}{\sqrt{p^2+q^2}}$$ I understand the proof of this formula ...
Shubhraneel Pal's user avatar
51 votes
3 answers
45k views

What is the analogue of spherical coordinates in $n$-dimensions?

What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by $$x_1=...
a06e's user avatar
  • 6,729
23 votes
2 answers
7k views

Why does partial differentiation give centre of a conic?

Definition of centre of conics states that, "The point which bisects every chord of the conic passing through it is called centre of the conic" General equation of second degree is $S:ax^2+2hxy+by^2+...
MathGeek's user avatar
  • 1,357
9 votes
2 answers
9k views

A good Open Source book on Analytic Geometry?

Hi my course specifically talks about : Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, ...
0 votes
1 answer
895 views

Proving the Apollonian circle formula

Proposition: Let: $c,d\in\mathbb{C} \space : c\neq d$ Then: $\forall k \in \mathbb{R}^{+}\setminus\{1\} \forall z \in \mathbb{C}: |z-c|=k|z-d|$ represents Appolonius circle. What was already done: ...
user avatar
47 votes
7 answers
433k views

How to calculate the intersection of two planes?

How to calculate the intersection of two planes ? These are the planes and the result is gonna be a line in $\Bbb R^3$: $x + 2y + z - 1 = 0$ $2x + 3y - 2z + 2 = 0$
user1111261's user avatar
  • 1,149
13 votes
1 answer
6k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
Rob's user avatar
  • 133
6 votes
6 answers
27k views

Common tangent to two circles

Find the equations of the common tangents to the 2 circles: $$(x - 2)^2 + y^2 = 9$$ and $$(x - 5)^2 + (y - 4)^2 = 4.$$ I've tried to set the equation to be $y = ax+b$, substitute this into ...
JSCB's user avatar
  • 13.5k
1 vote
2 answers
6k views

The line $y=mx+c$ is a tangent to $x^2+y^2=a^2$ if:

The line $y=mx+c$ is a tangent to $x^2+y^2=a^2$ if: $1$. $c=a\sqrt {1+m^2}$ $2$. $c=\pm a\sqrt {1+m^2}$ $3$. $c^2=\pm a\sqrt {1+m^2}$ $4$. $\textrm {None}$ My Attempt: The tangent to circle $x^2+...
pi-π's user avatar
  • 7,426
37 votes
5 answers
89k views

What is the equation of a general circle in 3-D space?

I know that $(x-x_0)^2+(y-y_0)^2-r^2=0$ is a general planar circle and $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2-r^2=0$ is a general sphere. I want to know the general expression of a circle in space. Can ...
Hamid Farzane's user avatar
31 votes
4 answers
8k views

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables $\{x_1,...
Georgy's user avatar
  • 1,467
10 votes
9 answers
7k views

Distance Between A Point And A Line

Any Hint on proving that the distance between the point $(x_{1},y_{1})$ and the line $Ax + By + C = 0$ is, $$\text{Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ What do ...
alok's user avatar
  • 3,910
1 vote
2 answers
281 views

Finding the equations of the lines and tangent to the circle

Find the equations of the lines through $(2,0)$ and tangent to the circle $x^2+y^2=1$. I tried to solve this and I know the right answer but I just can't solve this. The right answer: $\sqrt{3}y=x-2$ ...
user146411's user avatar
24 votes
8 answers
61k views

Determining if an arbitrary point lies inside a triangle defined by three points?

Is there an algorithm available to determine if a point P lies inside a triangle ABC defined as three points A, B, and C? (The three line segments of the triangle can be determined as well as the ...
Casey's user avatar
  • 409
12 votes
2 answers
33k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
Alessandro Jacopson's user avatar
6 votes
5 answers
2k views

Nature and number of solutions to $xy=x+y$

Find all solutions to $$xy=x+y$$ Initially the given condition was $x,y\in \Bbb{Z}$. $$$$In this case, I just guessed that the solutions were $(0,0)$ and $(2,2)$. As far as I can see, these are the ...
user342209's user avatar
39 votes
14 answers
57k views

Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the ...
Justin L.'s user avatar
  • 14.6k
21 votes
3 answers
22k views

The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
ReekMaths's user avatar
  • 821
8 votes
2 answers
6k views

Proof of Descartes' theorem

I came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any ...
rock321987's user avatar
7 votes
5 answers
54k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
help's user avatar
  • 119
5 votes
2 answers
43k views

What is condition for second degree equation to represent a pair of straight lines?

According to my text the necessary and sufficient condition for a general equation of second degree i.e. $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a pair of straight lines is that 1) the ...
Matt's user avatar
  • 1,160
4 votes
1 answer
1k views

Showing positive definiteness in the projection of ellipsoid

I would like to show that the projection onto the $xoy$ plane of the centered ellipsoid given by the definition $$\mathbf{x'}\mathbf{A}\mathbf{x}=1$$ where we have a positive definite $$\mathbf{A}= \...
Mathemagical's user avatar
  • 3,519
32 votes
4 answers
14k views

What Does Homogenisation Of An Equation Actually Mean?

For example, if we have a conic: $$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$ What does homogenising this equation with another line (say $ax + by + c = 0$ ) actually mean? As in, what are the ...
Anna's user avatar
  • 455
15 votes
3 answers
11k views

What is the difference between vector components and its coordinates?

Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is ...
Val's user avatar
  • 1,481
12 votes
4 answers
2k views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
hmakholm left over Monica's user avatar
12 votes
3 answers
3k views

How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$ a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3 $$ then the vertices are actually ...
Hogarth's user avatar
  • 123
8 votes
3 answers
1k 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 ...
Han de Bruijn's user avatar
3 votes
3 answers
34k views

How to find shortest distance between two skew lines in 3D?

If given 2 lines $\alpha$ and $\beta$, that are created by 2 points: A and B 2 plane intersection I want to find shortest distance between them. $$\left\{\begin{array}{c} P_1=x_1X+y_1Y+z_1Z+C=0 \\...
Margus's user avatar
  • 159
3 votes
1 answer
3k views

How to obtain the equation of the projection/shadow of an ellipsoid into 2D plane?

Given an ellipsoid equation of the form \begin{equation}\label{eq_1}x'Ax=1\end{equation} where $A\in\mathbb{R}^{n\times n}$ is positive definite and non-diagonal and $x\in\mathbb{R}^n$. So, how can I ...
Jairo Giraldo's user avatar
2 votes
1 answer
847 views

Is there geometric proof for the equation of hyperbola using only constant distance from two foci definition?

Is there geometric or visual (but rigorous) proof for the equation of hyperbola whose foci are on the $x$-axis using only the traditional definition of hyperbola: a hyperbola is a set of points such ...
1b3b's user avatar
  • 1,276
32 votes
7 answers
5k views

Why do we believe the equation $ax+by+c=0$ represents a line?

I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form $ax+by+c=0$. I'm wondering why do we "believe"...
DancefloorTsunderella's user avatar
18 votes
8 answers
133k views

How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} ...
Casper's user avatar
  • 1,049
17 votes
3 answers
21k views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
Martin Sleziak's user avatar
16 votes
1 answer
16k views

How does the homogenization of a curve using a given line work?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
Tesla's user avatar
  • 2,136
11 votes
7 answers
895 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
Semiclassical's user avatar
6 votes
6 answers
6k views

How can I calculate a $4\times 4$ rotation matrix to match a 4d direction vector?

I have two 4d vectors, and need to calculate a $4\times 4$ rotation matrix to point from one to the other. edit - I'm getting an idea of how to do it conceptually: find the plane in which the vectors ...
Reykjavik's user avatar
5 votes
6 answers
16k views

Analytic Geometry (high school): Why is the sum of the distances from any point of the ellipse to the two foci the major axis?

I don't understand where that formula came from. Could someone explain? For example any point $(x,y)$ on the ellipse from the two foci $(-c,0)$ and $(c,0)$ is equal to $2a$ where $2a$ is the distance ...
Person's user avatar
  • 928
5 votes
5 answers
9k views

Find all triangles of which perimeter and area are numerically equal

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
Satvik Mashkaria's user avatar
4 votes
1 answer
964 views

How to create an Ellipse using a Light Source with a Conical Beam

A light source with a right circular conical beam is placed at a height $h$ above the origin of the Cartesian $x$-$y$ plane (i.e. positioned at $(0,0,h)$), and directed vertically downwards. The ...
Hypergeometricx's user avatar
71 votes
7 answers
6k views

How to straighten a parabola?

Consider the function $f(x)=a_0x^2$ for some $a_0\in \mathbb{R}^+$. Take $x_0\in\mathbb{R}^+$ so that the arc length $L$ between $(0,0)$ and $(x_0,f(x_0))$ is fixed. Given a different arbitrary $a_1$, ...
sam wolfe's user avatar
  • 3,417
27 votes
7 answers
139k views

How to find coordinates of reflected point?

How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? Example Question: If $P$ is a reflection (image) of point $(3, -3)$ in the line $2y = x+1$,...
Happytreat's user avatar
23 votes
4 answers
4k views

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
bubba's user avatar
  • 43.6k
22 votes
2 answers
142k views

Equation of a plane passing through 3 points

It should be simple, but I'm having trouble. The three points are $$A(1,-2,1)\qquad B(4,-2,-2)\qquad C(4,1,4)$$ The plane I get is $$x+2y+z+6=0$$ but it obviously does not pass through the three ...
blundered_bishop's user avatar
20 votes
5 answers
15k views

How is the angle between 2 vectors in more than 3 dimensions defined?

I would like to know how the angle between two n-vectors is defined. I mean whether it is unique and how we may compute it (is the inner product a valid method in the n-dimensional space?). I have ...
John's user avatar
  • 201

1
2 3 4 5
14