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

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2
votes
2answers
116 views

Find out for which values of $\lambda$ the points of the line are inside the circle

We have a line (in parameter): $ x = 2\lambda $ $ y = 1-\lambda$ Find out for which values of $\lambda$ the points of the line are inside the circle of $x^2+4x+y^2-6y+5=0$ What I did: I rewrote ...
0
votes
3answers
58 views

Find the lines which have a certain distance from a certain point

We have a point $P(1,7)$, get the equations of the 2 lines which have a distance of $5$ from point $P$. Both of the lines go through the origin. So I used the formula ...
3
votes
2answers
236 views

Geometry - Area of Siamese Triangles

How can I find the Area of this figure? It is quite curious because it is a particular case of this sequence: Anyone know how to find the area of this sequence as a function of the number of ...
0
votes
1answer
182 views

Find all of the circles which are tangent to another circle

We have the circle $(x-a)^2 + (y-a)^2 = a^2$ (which always is tangent to both of the axes). There are 4 of these circles which are tangent to the circle $x^2+y^2=2$. Get the 2 positive values for $a$ ...
0
votes
2answers
594 views

Problem with finding the equations of the lines tangent to a certain circle

This is a long question, and might seem like a repost of my earlier questions, but it isn't, hear me out: In my book is written: The equation of the line tangent to the circle $x^2+y^2=r^2$ in the ...
0
votes
5answers
723 views

Find the equation of a circle which intersects another circle perpendicularly

'Find the equation of the circle with its center at $M(4,3)$ which intersects the circle $(x-3)^2+y^2=5$ perpendicularly' How can 2 circles have a perpendicular intersection, is this even possible? ...
-1
votes
2answers
2k views

Urgent - Find the equation of the lines tangent to a circle

Question: 'Find the equation of the lines from point $P(0,6)$ tangent to the circle $x^2+y^2=4x+4$. So what I did firstly is rewrite it to the form $(x-2)^2 + y^2 = 8$, and I saw that point $P$ is ...
1
vote
0answers
49 views

Solving inverse square of visible scale

I'm not super-adept in mathematics, so I turn to you for help. As I read, the perceived scale of an object reduces by the inverse square as the viewed distance increases. In order to solve for this, ...
3
votes
0answers
81 views

Algebraic Geometry studied via Filters

Is there any research relating varieties with filters instead of radical ideals? For example, Suppose we have a variety V in C^n, now consider the fixed filter consisting of all algebraic sets ...
1
vote
2answers
2k views

Can you suggest me a good book for self-study of analytic geometry 1?

I'm stuyding mathematics alone, but I plan to enter in the university in the near future, I went to the university website and it suggests Analytic Geometry 1 as part of their curricula, the topics ...
2
votes
2answers
350 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, ...
2
votes
2answers
489 views

How to define a perspective circle in xy?

You can see a perspective view of a square(FCED) and a circle in 2D screen. O is center of the circle. How can I define the perspective circle equation that shown as red in the picture? Thanks a ...
1
vote
2answers
501 views

The intersection of a line with a circle

Get the intersections of the line $y=x+2$ with the circle $x^2+y^2=10$ What I did: $y^2=10-x^2$ $y=\sqrt{10-x^2}$ or $y=-\sqrt{10-x^2}$ $ x+ 2 = y=\sqrt{10-x^2}$ If you continue, $x=-3$ or ...
1
vote
1answer
954 views

Analytic Geometry: Circle

Given is a line with parametric equation: $ x = 2 \lambda $ $ y = 1-\lambda $ Find out for which values of $\lambda$ the line is inside the circle of $x^2+4x+y^2-6x+5=0$ My attempt at solving ...
6
votes
5answers
17k 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
1k 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) $ ...
3
votes
4answers
8k 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
189 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
229 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
200 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
167 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?
1
vote
6answers
12k 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 ...
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
496 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
123 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
185 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
248 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
51 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
89 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
41 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
349 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
79 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
52 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 ...
0
votes
1answer
85 views

Getting coordinate between two coordinates knowing the distance and latitude

That is my wall: I know the coordinates of the lower points (left and right). (X1,Y1,Z) and (X2,Y2,Z) where X is the latitude, Y longitude and Z the altitude. I want to know the another point of ...
2
votes
1answer
218 views

Intersection of 3D curves parameterised by piecewise defined functions

I need to calculate the intersection of two 3D parametric curves $\vec{C_1}$ and $\vec{C_2}$. Those curves are parameterised by piecewise functions. $\vec{C_1}= ...
5
votes
4answers
20k 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
99 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
570 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
98 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
49 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
71 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
360 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
52 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
207 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, ...
1
vote
3answers
339 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
92 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
7
votes
7answers
1k 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
78 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
1k 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
116 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 ...