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

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0
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2answers
21 views

Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
3
votes
1answer
395 views

$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point ...
0
votes
1answer
15 views

How to points of line in ellipse when it's moved as ellipse tangent [on hold]

Ellipse Picture In the picture minor and minor asymptote and points of line (X&Y) are known, when we move the line new position is X' and Y'. How can be calculated new position of line or is ...
0
votes
1answer
25 views

A specific case of quadratic forms

I have a quadric as follows: $$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?
1
vote
1answer
7 views

Lines intersections distance on the asymptotes

Like in picture we have two lines. Lenght of one of them is 2E and other's lenght 2C and also ellipse asymptotes are A and B and its center is on origin(0,0) I want to find D and F How can I ...
0
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1answer
511 views

deriving formula for reflection over y=mx+b using dot product

So, I know that the formula for a generic point is $$\left(\frac{1-m^2}{1+m^2}x + \frac{2m}{1+m^2}(y-b), \left(\frac{2m}{1+m^2}\right)x - \left(\frac{1-m^2}{1+m^2}\right)(y-b)+b\right)$$ when you ...
0
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0answers
6 views

Angle of the tangent vector of a parabola in function of the angle of the vector that defines this parabola (Apostol, chapter 14.21, problem 1)

Apostol, chapter 14.21, problem 1 (a review problem) Here is the question: Let r denote the vector from the origin to an arbitrary point on the parabola $y^2 = x$, let $\alpha$ be the angle that ...
7
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6answers
2k views

Where can I find Linear Algebra in Nature?

I'm a Computer Science major and I've been studying Analytic Geometry and Linear Algebra this semester. Today my teacher gave a hell of an explanation talking about linear systems, quadratic ...
1
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0answers
48 views

4 points in 3-d space (one known and three unknown)

Problem in 3-d space. We have four points: $P_0$ where we know coordinates $(0,0,0)$ and $P_1, P_2, P_3$ where coordinates are unknown. However we know distances between $P_1, P_2, P_3$ (let's name ...
0
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0answers
11 views

Same Center Ellipse Major and Minor Axes

Ellipse Picture I have two same center ellipses A, B, and C are known values X and Y arent known values and I need to obtain these values. How can it be calculated?
3
votes
1answer
441 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, ...
-1
votes
2answers
24 views

Calculate the plane equation of 2 vectors. [on hold]

Which type should I use in order to calculate the plane equation that is defined by 2 vectors, let's say V1 $\langle{1,2,3}\rangle$ V2 $\langle{4,5,6}\rangle$.
0
votes
0answers
25 views

How to find the Coordinate equation of a curve which bends all the parallel rays from infinity towards a single point

How should I proceed on to find the coordinate equation of a curve such that it bends all the parallel rays coming from infinity towards a single point. Yes I know that it would be a 2nd degree ...
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0answers
11 views

Мöbius Transformations and circle inversion

Can a Möbius Transformation be decomposed into a composition of 2 generalized circle inversions?
1
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1answer
19 views

Proof of the reflective property of the ellipese

I'm trying to prove the reflection property of the ellipses for an optics problem. The property is that that a ray of light originated at one of the ellipse's foci reflects in such a way to pass ...
1
vote
4answers
46 views

Is this a correct way to solve this high school coordinate geometry question?

Here's the question: Given point $A$: $(-3;-1)$ Given point $B$: $(3;7)$ Given point $Z$: $(x;0)$ Find the $x$ coordinate of point $Z$ so that the angle of view of AB segment is $90$ ...
0
votes
1answer
92 views

Plotting 3 equidistant points on a sphere

I'm trying to figure out how to plot with $x,y,z$, three points that are equidistant along the surface of a sphere from each other that are all on a horizontal axis (so $y = 0$) with a radius of $500$ ...
0
votes
1answer
32 views

Getting the coordinates of the center of a circle bisecting two other circles.

We have circles $C_1$ and $C_2$ with centers $(-d,0)$ and $(d,0)$, radii $a_1<d$ and $a_2<d$ respectively. If circle $D$ with radius $r$ (and with centre not necessarily on the x-axis) bisects ...
0
votes
1answer
14 views

Finding 3rd circle's coordinate of particular radius given 2 circles coordinate, circles touch externally

Given circle say A,B,C where each of them touches each other externally . We are given radius of all 3 circles. We are also given 2-D coordinates of centre of B,C ,we need to compute coordinates of A. ...
2
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0answers
39 views

Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a $2$-D figure has maximum distance from a fixed coordinate. Let me demonstrate: $3$ points: $(1,3), (2,2), (5,0) $; Fixed ...
5
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2answers
63 views

Different methods for finding the minimum of $|x-2y|$ when $x^2+1=2y^2$.

For $x, y \in \Bbb R$, $x^2 + 1 = 2y^2$, find the minimum of $|x - 2y|$. At a glance I found that the point $(x, y)$ lies on a hyperbola and $|x - 2y|$ is just the distance between the point and the ...
0
votes
1answer
672 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
2
votes
4answers
1k views

Intersection of ellipse with circle

I would like know whether a circle is intersecting an ellipse. Here ellipse equation is $$Ax^2 + Bxy + Cy^2 + dx+ey + 1 = 0,$$ and the circle equation is $$(x-g)^2 + (y-f)^2= r^2.$$
4
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1answer
34 views

the area that a part of an ellipse consumes in a square of a discrete grid

Think about a discrete grid of unit 1, which means the grid consists of infinite number of squares whose area is 1. You can assign a coordinate to each square and one of them will have the coordinate ...
0
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0answers
14 views

Linear functionals and hyperplanes

If $L:\Bbb R^n\to\Bbb R$ is a non-trivial linear functional , i.e $L(x+y)=L(x)+L(y), x,y \in\Bbb R^n$ and $L(ax)=aL(x), x \in\Bbb R^n, a \in\Bbb R$, then why does the set of all x $\in\Bbb R^n$ that ...
1
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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 ...
0
votes
0answers
37 views

Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
2
votes
0answers
42 views

Given any parametric curve, finding its general form?

I'll illustrate the problem I'm trying to solve with an example. Let's consider the equations $$ x = \cos (t) $$ $$ y = \sin (t) $$ We know that these are a parametric form of the unit circle. In ...
5
votes
3answers
274 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 ...
2
votes
1answer
53 views

Plot points on an arc

I have modified this post with updated information so the problem may be more clear. Because the answer provided does not achieve the results intended, maybe adjusting the content will help adjust ...
1
vote
1answer
34 views

Formula for area of triangle in complex plane [closed]

If $A(z_1)$, $B(z_2)$, $C(z_3)$ are vertices of a triangle $ABC$ in Argand plane, what is the area of the triangle?
-4
votes
1answer
36 views

pls help to the sum [closed]

A plane passes through a fixed point $(a,b,c)$. Show that the locus of the foot of perpendicular to it from origin is the sphere $x^2+y^2+z^2-ax-by-cz=0$
0
votes
1answer
27 views

Find the point on the plane xOy [closed]

Let $A(x_1; y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$ be three points not lying on the same straight line. Find the point on the plane $xOy$ such that the sum of the distances from it to these points is ...
5
votes
2answers
283 views

Geometric interpretation for eigenvalues and eigenvectors of the cross product's representation as a linear map

Fix ${\bf x} = (x_1,x_2,x_3) \in \Bbb R^3\setminus\{{\bf 0}\}$. We can look at the cross product as a linear map ${\bf x}\times: \Bbb R^3 \to \Bbb R^3$ which is represented in the standard basis by ...
0
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2answers
36 views

Proof that if two lines are parallel then $A_1$ = $A_2$ and $B_1$ = $B_2$?

Let two lines to be parallel in their general form. $L_1$ : $A_1 x$ + $B_1 y$ + $C_1$ $L_2$ : $A_2 x$ + $B_2 y$ + $C_2$ Now i wish to prove $A_1$ = $A_2$ and $B_1$ = $B_2$ But i can only think of ...
0
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0answers
22 views

Find a linear sequence of the common terms between 2 other sequences?

Given 2 linear sequences $an_1+b$ and $cn_2+d$ generate a sequence of whole numbers that can expressed as $an_1+b$ and $cn_2 + d$. An example to illustrate this: Given $2n+3$ and $3n+6$ the sequence ...
2
votes
3answers
1k views

given coordinates of beginning and end of two intersecting line segments how do I find coordinates of their intersection?

There are two line segments. I know for sure they intersect (so I don't have to check it). For both line segment I know coordinates of its both ends. With what formula can I find coordinates of their ...
5
votes
3answers
136 views

Finding equation of chord of hyperbola.

Equation of chord of hyperbola joining points $(a\sec\phi,b\tan\phi)$ and $(a\sec\phi_1,b\tan\phi_1) $ $$y-b\tan\phi=\frac{b\tan\phi-b\tan\phi_1}{a\sec\phi-asec \phi_1}(x-a\sec\phi) $$ This reduces ...
2
votes
2answers
29 views

Slope of axes of a General Conic Section

A General Conic Section is given by the equation $ax^2 + by^2 + 2hxy +2gx +2fy + c =0 $. Let the $\theta$ be the slope of one of its axes. Prove that : $$\tan 2\theta = ...
0
votes
1answer
21 views

Hyperbolas and Quadrants on Rotation

Let's assume we have a standard hyperbola. On rotating the hyperbola $45^{\circ}$ clockwise, the new hyperbola should lie in the $2$nd and $4$th quadrant. However, the equation of a parabola rotated ...
0
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2answers
69 views

Geometrical interpretation of solving a $3 \times 3$ system of equations

Solve the following system of equations and give a geometrical interpretation of the result. \begin{align*} x + y + z &= 6\\ 2x + y − 3z &= -5\\ 4x − 5y + z &= −3 \end{align*} I know that ...
0
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1answer
31 views

Coordinates of incentre without finding side lengths

If I am given the equations of sides of a triangle and I need to find incentre what is the shortest method ? Is it possible without having to find lengths of sides of triangle?
1
vote
1answer
32 views

Vectors: Using Pythagoras's theorem for magnitude in the 4th dimension

For a simple x and y plane (2 dimensional), to find the distance between two points we would use the formula $$ a^2 +b^2 = c^2 $$ For a slightly more complicated plane; x,y and z (3 dimensional), ...
3
votes
0answers
34 views

Cosine Inequality

Show that given three angles $A,B,C\ge0$ with $A+B+C=2\pi$ and any positive numbers $a,b,c$ we have $$bc\cos A + ca \cos B + ab \cos C \ge -\frac {a^2+b^2+c^2}{2}$$ This problem was given in the ...
1
vote
2answers
20 views

Suppose you graphed every single point of the form (2t + 3, 3-3t).

Suppose you graphed every single point of the form $(2t + 3, 3-3t)$. For example, when $t=2$, we have $2t + 3 = 7$ and $3-3t = -3$, so $(7,-3)$ is on the graph. Explain why the graph is a line, and ...
2
votes
2answers
23 views

The expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$.

Using vectors I tried obtain the expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$. The point of intersection is ...
0
votes
0answers
24 views

Homogenization of Equations

Say there are two equations: $3x^2+mxy-4x+1=0$ and $2x+y-1=0$. I have to find possible values of $m$ for which lines joining the points of intersection of above two equations are at right angles. I ...
0
votes
1answer
12 views

Points on parabola with abscissa in A.P. and ordinate in G.P.

The points with coordinates $(a,b),(a_1,b_1),(a_2,b_2)$ are points on parabola $y=3x^2$. The numbers $a,a_1,a_2$ are in Arithmetic progression while $b,b_1,b_2$ are in Geometric Progression. Calculate ...
2
votes
1answer
18 views

General conic equation and coefficient matrices

For a general conic $Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ we define a matrix $A$ as follows: $A=\left( \begin{matrix} a& h& g\\ h& b& f\\ g& f& c\end{matrix} \right)$. Then we ...
5
votes
7answers
22k views

Find if three points in 3-dimensional space are collinear

Find if the points joining $A=(6,7,1), B=(2,-3,1)$ and $C=(4,-5,0)$ are collinear. How to determine collinearity in three dimensions? In two dimensions, one can compare the slopes of segments ...