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

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2
votes
4answers
11k 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?
0
votes
4answers
908 views

Analytic Geometry: Distance between a point and a line.

Get the equations of both lines going through $0$ which have a distance of 5 from the point $(1,7)$. How to handle this problem? We have this formula: If line $l$ is $ax+by=c$, distance $ P(x,y) $ ...
2
votes
4answers
5k 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 ...
0
votes
2answers
132 views

Problem involving points on a line.

Given two points A(-3,4) and B(2,5) find the coordinates of one point P on the line and passing por A and B. Look that the point P is two times more distant from A than B.
1
vote
1answer
179 views

Finding two points that have a defined distance between two intersecting lines

I was given a test yesterday, a test which unfortunately I was unable to study to. In it was a question that was too hard that it became our homework for the whole week. It says that on lines ...
3
votes
1answer
139 views

Offsetting a curve in 2D

I'd like to "move" a curve $d$ (offset) units "up" (actually in the sense that the perpendicular distance between the curves is always constant). The objective is to create a channel that has constant ...
1
vote
3answers
154 views

Number of ellipses through two fixed points in 2D space?

How many ellipses with a given size (mean $a$ and $b$ given) one can draw through two fixed points in 2D plane?
0
votes
0answers
89 views

Please help with this vector question

The points A, B and C have position vectors a=(2,1,2) b=(-3,2,5) and c=(4,5,-2) respectively, with respect to a fixed origin. The point D is such that ABCD, in >that order, is a parallelogram. ...
0
votes
3answers
5k views

Find if the points are collinear

Find if the points joining $A=(6,7,1), B=(2,-3,1)$ and $C=(4,-5,0)$ are collinear. how to prove that? Anyone, please help me!
0
votes
3answers
1k views

Show by using vectors that the two diagonals of a square are equal in dimension

and also perpendicular to each other? how can we prove that ...please Help me
2
votes
2answers
311 views

Pattern matching circle, square or triangle

I have a set of x, y co-ordinates that are actually taken from hand drawings of a circle, square or a triangle. Using the set of points, I need to mathematically find if the points approximately fit a ...
0
votes
2answers
95 views

What are the coordinates of vertex C

Triangle $ABC$ has 2 given vertices, $A(1,1)$ and $B(5,3)$. Also, AC=BC and $\angle ACB = \,^{\circ}\mathrm{90}$. The triangle is in the first quadrant entirely. What are the coordinates of vertex ...
2
votes
2answers
176 views

Analytic Geometry question I can't solve

An isosceles triangle $ABC$ has 2 given vertices, $A(3,2)$ and $C (7,14$). The slope of AB is $\dfrac{1}{2}$. What are the coordinates of B? I could figure out that line AB = $\dfrac{1}{2}x + ...
0
votes
1answer
168 views

Find the coordinates of P

There is a point P on the line $5x-3y=7$ that is equally far from the points $A(1,4) $ and $ B(3,10)$. Find the coordinates of P. What I did: $5x-3y=7$ is the same as $ y = ...
0
votes
6answers
48 views

Finding the slope of a function

How do I find the slope of this function: $px + (2p-1)y + 4 = 0$ I need to know how to answer a previous question of mine (also posted on this forum)
0
votes
5answers
79 views

Determine a parameter in such a way that two lines are parallel

The lines $px + (2p-1)y + 4 = 0$ and $(p+3)x + 2py + 6 = 0$ are parallel to each other. Find $p$. I have no idea how to tackle this problem, can anyone help?
1
vote
1answer
39 views

How to figure out x and y components of a vector

Given a vector $\vec{v}$ and an angle $\alpha$ between the vector and $Oy$ or $Ox$ what is the quickest way to figure out the projections, or the $x$ and $y$ components?
4
votes
2answers
260 views

Locus of points generates several very different curves. Closed form?

Consider, for the sake of simplicity, a circle $C$ centered at he origin with radius $a$. Let $F=(h,k)$ be a point not necessarily inside the circle. Let $M=(a\cos\theta,a\sin\theta)$ be a point in ...
1
vote
1answer
76 views

Locus perpendicular to a plane in $\mathcal{R}^4$

I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me.. In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the ...
1
vote
0answers
51 views

figuring out the sum of angles from the addition of geometrical shapes within a circle

if you add a number with another you get the sum of the two.(1+1=2, 2+2=4) right? but if you take a circle and put a horizontal line through it you have 4 angles in it. put another circle with a ...
4
votes
4answers
9k views

How to know if a point is inside a circle?

Having a circle with the centre $(x_c, y_c)$ with the radius $r$ how to know whether a point $(x_p, y_p)$ is inside the circle?
2
votes
1answer
96 views

Hard calculating about bisectors

I need to find intersection point of two bisectors of KL, KC and CL line. Coordinates : $C(2a+2, 2b), L(2-2a, 2b)$ and $K \left( \frac{2(a^3 - a^2 + ab^2 + b^2)}{a^2 + b^2}, 2 - \frac{4a}{a^2 + ...
0
votes
1answer
505 views

intersection point between circle and line

The line $-bx + ay + 2b = 0$ intersects circle on points A and B. Circle equation is $$(x-1)^2 + \big(y-\frac{a^2 + b^2 - a}{b}\big)^2 = \frac{(a^2 + b^2 - a)^2 + b^2}{b}$$ or after ...
2
votes
3answers
97 views

Simple analytic geometry question I need help with

Give the equation of a circle with the center $ (a,0) $ which is tangent to the line $ y = x $ I now have $ (x-a)^2 + y^2 = r^2 $ but I don't know how to continue.. please help!
2
votes
2answers
44 views

Trouble with formulation of an analytic geometry question

I'm having trouble understanding a certain question, so I am asking for an explanation of it. The question is asked in a different language, so my translation will probably be mistake-ridden, I hope ...
0
votes
3answers
70 views

Mathematical 'language' (geometry)

What does this question mean: 'Show (translated from my native language) that the equation $ x^2 - 4x + y^2 + 6y = 51 $ is a circle.' I have absolutely no idea how to 'show/prove/etc.' it, other than ...
1
vote
3answers
238 views

How to get the equation of a circle with the given information

You have a center of a circle $M(-2,3)$ going through a point $P(1,7)$. What is the equation of the circle? I thought you could solve it by $R^2 = 3^2 + 4^2$, but that would just give a radius of 5 ...
1
vote
2answers
50 views

An analytic geometry question + algebra

We have a Cartesian coordinate system with the points M (a,b) Q (4,2) and P (x,y) but I don't think you need P to solve this one, only M and Q. M is the middle of a circle with a radius r, and Q is a ...
0
votes
0answers
167 views

Analytic Geometry Question

There is a point $M (2,1$) on a Cartesian coordinate system. There is also a point $P (x,y)$. What is the distance $MP$ in $x$ and $y$? I can figure out that $ PM^2$ = $(x-2)^2 + (y-1)^2$, at least, ...
0
votes
3answers
237 views

How to find on which outer side of the rectangle falls the point?

Qt has a class QRect which tells whether the point is inside the rectangle or not. Now, the problem is to find out on which ...
0
votes
0answers
79 views

Eccentricity and Length of Semi Axes of a conic

If a conic $ax^2+by^2+2hxy+2gx+2fy+c=0$ and say: How to find the eccentricity and the semi-axes of this conic. I do understand that if its a hyperbola only one of the semi axes will be real. Soham
6
votes
7answers
938 views

Detect when a point belongs to a bounding box with distances

I have a box with known bounding coordinates (latitudes and longitudes): latN, latS, lonW, lonE. I have a mystery point P with ...
3
votes
0answers
73 views

packing for the polytope

Let $X=(X, \|\cdot\|)$ be some normed space. Let $C=[-1,1]^n$ and $H$ be a plane with equation $\sum_{i=1}^nr_i=s, 1\le s\le n.$ (Here $r_i$ are such that $Proba(r_i=1)=Proba(r_i=-1)=1/2$). The ...
5
votes
3answers
805 views

A simple(?) Analytical Geometry Question (Ellipse) my teacher can't solve

Here's the story: I am a high school student who absolutely loves math. So I took a university level mathematics course that is renowned throughout our school for being extremely rigorous and tough. ...
0
votes
1answer
114 views

Finding coordinates of some points in picture

Let $A(0,0)$, $B(2,0)$, $C(c_{x}, x_{y})$, $D(d_{x}, d_{y})$. $O_1$ and $N$ is the center of circles (ABD) and (CKL). Find coordinates of $C, N, K, L, O_1$.![Don't take care about coordinate system in ...
1
vote
1answer
142 views

Parametrization of a solid

Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace $.
1
vote
2answers
38 views

How do I solve a function with x^2 and x^-1 to x?

We got two functions: $f(x)=ax^2+b$ $g(x)=x^{-1}=1/x$ I know that they are touching each other in $x=1$. Now I can find out the values for $a$ and $b$ in $f(x)$. Set the derivative of both ...
3
votes
0answers
131 views

Illustrations of a line and a curve intersecting for complex field

Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
0
votes
1answer
123 views

locate a audio source by 3 microphones, same plane, by using the volume

Let have 3 microphones MIC1: @ mic1x,mic1y MIC2: @ mic2x,mic2y MIC3: @ mic3x,mic3y. MIC1-MIC2 separated 2500mm at 90degr MIC2-MIC3 separated 2500mm at 90degr If a sound produced by source S reach ...
79
votes
20answers
19k 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?
4
votes
3answers
801 views

Find a plane perpendicular to a plane passing by point

In $\mathbb R^4$ I have: $$\pi: \begin{cases} x+y-z+q+1=0 \\ 2x+3y+z-3q=0\end{cases}$$ I have to find $\pi' \bot$ $ \pi $ and passing by $P=(0,1,0,1)$. How can I do that? Thanks a lot!
2
votes
1answer
155 views

How to constrain disks that intersection of them is inside unit circle

I have two disks $(x-a_1)^2+(y-b_1)^2\leq r_1^2$ and $(x-a_2)^2+(y-b_2)^2\leq r_2^2$, where $a_1$, $b_1$, $r_1$, $a_2$, $b_2$, $r_2$ are all known. What kind of constraint can I put on $a_i$, $b_i$ ...
2
votes
1answer
143 views

Showing: point of polytope which maximizes the minimum distance to a vertex is a barycentre?

Let $T_1$ and $T_2$ be two regular $(n-1)$-dimensional simplices with vertices $$(t,0,\ldots,0), (0,t,\ldots, 0),\ldots, (0, 0, \ldots, t),$$ and $$(t-n+1,1,\ldots, 1), (1, t-n+1, \ldots, 1), \ldots, ...
1
vote
1answer
280 views

How to find intersection of an ellipse and a line that passes through the foci

There are two lines, parallel to the $x$-axis, which pass through the foci and intersect the ellipse at four points. How can I find the points of intersection? vertex: $(0,0)$ foci: $(0,10)$ and ...
1
vote
2answers
1k views

Mathematics behind intersection points of two lines using quadratic equation

This is the question I am trying to solve. I do not need any code examples just help on mathematics. Suppose two line segments intersect. The two endpoints for the first line segment are $(x_1, ...
0
votes
1answer
869 views

How do I “describe the region” of $\Bbb R^3$ represented by an inequality?

I don't really understand what these two questions are asking. Nor do I know how to start it. 1) Describe in words the region of $\mathbb {R}^3$ represented by: $x^2 + z^2 \le 9$. 2) Write ...
2
votes
0answers
476 views

Equations of branches of a mind map

Sorry for the long question, but it's not so simple to explain. Consider a mind map like this: I want to draw branches in a cartesian coordinate system. I'd like to find two equations which ...
1
vote
3answers
4k views

Find area of a triangle given the equation of sides

Find the coordinates of the vertices of the triangle $ABC$ formed by the intersection of the lines $x + y = 0$, $x = 0$ and $y = x - 1$. Hence, find the area of the triangle. I tried to sketch a ...
1
vote
0answers
113 views

Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
1
vote
1answer
104 views

Finding a coordinate with no intermediate variables

I want to know if it possible, using only the $+ - \div \times$ operators to solve a simple geometry problem. The questions is further complicated because I want to integrate it into a very restricted ...