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

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2
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3answers
325 views

Parametrization of $y^2 - x^2=1$

I have found parametrizations for the level curve $y^2-x^2=1$, however, I have a question regarding one of them. From the Pythagorean trigonometric identity $\cos^2 x + \sin^2 x =1$ we obtain ...
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0answers
41 views

Equations for Intersection of Plane and circle

I am having a problem in getting the related equations for an intersection between a point on a plane and the edge of a circle in 3D space. Any suggestions? Or also a tangent plane and a point on a ...
1
vote
3answers
69 views

Analytic Geometry question, planes and lines

Let there be a plane going through three points $(0,2,-9), (0,-1,0), (-\frac{3}{m},1,-3)$. For which value of $m$ is the line $l: (3,0,-9)+t(2m,-5,7)$ onto (or 'inside') the plane? Not sure how to ...
2
votes
2answers
450 views

How to find vertex of a parabola from its second degree equation

Given a parabola with second degree equation as $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$ assume that this isn't degenerate case, and $B^2-4AC=0$ How can I find its vertex position?
3
votes
2answers
135 views

Equation of a circle given one point and two lines

Find the equation of the circle that pass through $(2,3)$ and are tangent to both the lines $3x - 4y = -1$ and $4x + 3y = 7$.
3
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2answers
95 views

Finding point on ellipse equally distant from two other points on the ellipse

I have an ellipse with two points on it: A and C (with known coordinates). Point O is the center of the ellipse (coordinates are given). I need to find coordinates of point B which also lies on the ...
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0answers
82 views

Volume of a regular tetrahedron vs Volume of a sphere

I have the following question: given a regular tetrahedron and a sphere that goes through the middle of all sides of the regular tetrahedron, which has a bigger volume? and what is the ratio of ...
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2answers
479 views

Circle touching the $y$-axis passing through two points

How to find the equation of the circle touching the $y$-axis given that it passes through two particular points?
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1answer
54 views

Plane equation question!

Could anyone explain me how to do tasks like this one: Plane is intersecting Oy axis when $y = 3$ and line equation is $ 2x + 4= y-2=z$ belonds to plane. Write plane equation.
2
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0answers
233 views

A sphere packing problem

Suppose there is a large sphere of radius $R$. We want to pack it with smaller spheres. The volume of the smaller spheres change depending on where they are situated in the larger sphere. A smaller ...
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1answer
1k views

A moving point has its distance from (1,3) always one-third of its distance from (8,2). Find the equation of its Locus.

A moving point has its distance from (1,3) always one-third of its distance from (8,2). Find the equation of its Locus. My equation displays a circle formed by the loci, I don't know if it's right. ...
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2answers
100 views

**Each Pair Bisects the angle between the other pair** then $pq=?$

If the pair of straight line $x^2-2pxy-y^2=0,x^2-2qxy-y^2=0$ are $\ni$ Each Pair Bisects the angle between the other pair then $pq=?$ I do not understand geometrically,mathematically the boldly ...
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1answer
196 views

Analytic geometry textbook introduction.

I need the help of others concerning a good-rigorous analytic geometry textbook. (high school level.) Thank you for the help!
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3answers
31 views

A question on co-ordinates of intersecting lines…Given in picture below

Please do also MENTION how you got the solution.........
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2answers
74 views

Analytic geometry simple question

need help in this. In a right triangle in the three-deminsion plane $ABC$, A=$(2,-3,4)$, B=$(1,-1,5)$. Find $C$ if its known $C$ is on the line $L: (1,5,-2)+t(3,0,-2)$. What I did was finding the ...
0
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1answer
29 views

Points defined by relations (an exercise from “System of Coordinates”)?

An exercise from "System of Coordinates" (by Gelfand, Glagoleva and Kirilov) asks me to "[t]ry to decide by yourself which sets of points are defined by these relations" and relations given are: a. ...
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2answers
1k views

To prove by vector method , non-parallel sides of a trapezium having equal diagonals are equal.

How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? (taking $ABCD$ to be the trapezium with ...
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0answers
55 views

Determine properties of a shape defined by a function?

I have a number of functions that define 3d shapes. They take a point $(x, y, z)$ as a parameter and return a real value representing information about the shape. On the surface of the shape: $f(x) ...
2
votes
2answers
222 views

Is there a real continuous function of a Cube?

Sorry if this seems like a basic question, but I'm having troubles finding an answer in my research. I'm looking for a function that if given a test point (x, y, z) will return a real value describing ...
3
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0answers
62 views

points possible on a circle

Let $A, B, C, D$ and $E$ be five points marked in clockwise order, on the unit circle in the plane (with centre at origin). Let $\alpha$ and $\beta$ be real numbers and set $f(p)=\alpha x+\beta y$ ...
4
votes
1answer
119 views

How do I make pi = 3?

This question emerges from a discussion on quora which concluded that if a circle was drawn on the surface of a sphere, the ratio of radius (from the circle's centre as projected to the sphere's ...
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2answers
457 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
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4answers
497 views

Direct method to find the equation of a circle.

Suppose we are given four concyclic points or two lines which intersect the axes in concyclic points. Many a times, one point has a variable as a co-ordinate. Suppose the concyclic points are ...
1
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1answer
56 views

Rays in fat parabolas

Let $\epsilon>0$. Let $F\subseteq \mathbb{R}^2$ be the set of all points that lie at a distance less than $\epsilon$ from the curve $y=x^2$. Can $F$ contain a ray? That is, is there (for some ...
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4answers
122 views

Given Two Lines, How to Find a Point on Line $2$ given a Specific Distance From Line $1$?

I have two lines (U and V). What is the method to calculate a point on V given a specified distance (d) from U? The lines may be assumed that they do intersect (are not parallel) and are straight ...
4
votes
3answers
188 views

Proof that a line cuts in half the area of a parallelogram iff it goes through the intersection of the diagonals?

I read a theorem in a book which says that a line bisects a parallelogram iff it goes through the intersection of the diagonals. The edge case of this is of course if the line is one of the diagonal ...
2
votes
2answers
69 views

Proving Coplanarity of 3 vectors

Let $a,b,c$ be three vectors such that $|a|=|b|=|c|=\sqrt{2} $ and $a\cdot b = b\cdot c = a\cdot c = -1 $ . How can I prove that they are all coplanar? I found that the angle between every two of ...
5
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1answer
176 views

Coordinate Transformations

I am physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) between coordinate transformation of two kinds : Rotation of coordinate (and ...
2
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3answers
110 views

prove that the rose (in the polar plane) has $2n$ “petals” when $n$ is even

prove that the rose $r=\cos(n\theta)$ (in the polar plane) has $2n$ "petals" when $n$ is even. How can I start this demonstration? I would appreciate your help
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0answers
165 views

How to calculate center coordinates of two reverse arcs in 3D space

Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? ...
2
votes
1answer
313 views

The formula for pitch circle diameter.

I want to put $n$ number of circle with $r$ radius each in a big circle. Want to calculate the radius $R$ of the big circle. How can this be achieved?
2
votes
1answer
69 views

Finding the locus of the following conditions

Find the locus of all points in the plane, whose distance from a constant point $F=(x_0,y_0)$ divided by their distance from the vertical line $L=\{(k,y)\mid y \in \mathbb R \}$ equals a constant ...
0
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1answer
74 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
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1answer
155 views

translate coordinates on circle to percentage?

I'm coming more from a programming point of view but the question is pure math. The only strange thing, I guess, is that the coordinate system is like this: ...
-1
votes
1answer
1k views

Equation of circle with given radius passing through two given points

Find the equation of the circles passing through two points on the $y$ axis at distances 3 units from the origin and having radius 5. (This a homework problem but I do not know how to solve it.)
0
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1answer
93 views

Find the equation of the hyperbola?

The hyperbola being an orthogonal parabola, for which $(-1,2)$ is a focal point and $x-y+1=0$ is an asymptote. If I have the equation for the asymptote $y=x+1$ is the center $(0,1)$? I do not know ...
0
votes
1answer
48 views

Find equation of hyperbola?

The hyperbola has center $(0,0)$, and goes through the points $(3,1)$ and $(9,5)$ and the coordinate axes are the symmetry axes. The correct answer is $x^2 - 3y^2=6$.
0
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1answer
104 views

General Coordinate Geometry Problem - How to deal with lines parallel to y - axis

In coordinate geometry, whenever we solve a problem we see that if the resulting solution is a line, then all the lines which are parallel to y - axis are left out since their slope will be $\infty$ ...
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2answers
53 views

An equation for an ellipse

Definition: An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point ...
2
votes
1answer
95 views

Unicursal Curve Double Points

To quote Goursat: It is shown in treatises on Analytic Geometry that every unicursal curve of degree n has $\frac{(n-1)(n-2)}{2}$ double points, and, conversely, that every curve of degree n ...
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1answer
142 views

Analytical Geometry problem with complex numbers - alternate solutions.

The question is to show that the equation of the lines making angles $45^\circ$ with the line: $$ \bar{a}z + a\bar{z} + b = 0; \;\;\;\;\; a,z \in \mathbb{C}, b \in \mathbb{R} $$ and passing through a ...
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votes
4answers
449 views

Can we plot a regular octagon on a set of axes, where all vertices of the octagon lie on integer co-ordinates?

I'm a high school teacher and someone asked me this in my class, and to be honest I'm quite stumped! I haven't done any high level math in such a long time, and I'm really not sure how to approach ...
1
vote
2answers
87 views

Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if ...
0
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3answers
131 views

2 dimensional coordinate geometry

If $L_1$ and $L_2$ are two lines belonging to the family of lines $(3+2s)x+(4+3s)y=7+5s$ such that they are at maximum and minimum distances from the center of the circle $3x^2 +3y^2 -12x-18y-91=0$, ...
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1answer
238 views

Show that the focus of the parabola lies on the nine point circle of the triangle.(dificult)

A parabola is drawn such that each vertex of a given triangle is the pole of the opposite side;show that the focus of the parabola lies on the nine point circle of the triangle and that the ...
2
votes
0answers
81 views

Clarification of the Jacobian

Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ ...
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1answer
169 views

Why is the Jacobian ${\rho}$

Just a little confused. When I find the volume of a cone (or a sphere) for that matter I multiply the partial derivatives by the Jacobian. ${\rho}$ for a cone. and ${\rho^2 \sin \phi}$ for a sphere. ...
2
votes
2answers
52 views

Proving the following is a Group

I'm studying this weird course called "Analytic Geometry", but in reality it seems like a mash of modern or abstract Algebra (...I'm not so sure...), and includes stuff like Affine transformations, ...
5
votes
2answers
3k views

Generalizing the hardest question on the practice math GRE

The most-missed question on the Math GRE is the following: How many times does $x^{12}$ intersect $e^x$? Because I told you it was hard, you probably realized it was a trick and got the right ...
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1answer
138 views

Trying to understand Volume of a cone without the unit sphere

I have been working on the double integral proof for the volume of a cone. I found that I can use a unit-sphere Where the base of the cone is the equator and the height is the distance ${\rho}$ to ...