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

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3
votes
2answers
59 views

Curve of Equal SWR

I'm trying to figure out how radio frequency "matching stubs" work. In order to fully understand the problem, I need to know how the "curve of equal SWR" looks like. I did a few plots, and it looks ...
19
votes
1answer
558 views

Intuition why the volume and surface area of the unit sphere eventually decrease

The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$ and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$ both attain maximum values for finite $n$. We can ...
0
votes
1answer
870 views

Help: Find the area of the shaded region

Given an arc PQ with curvature $\frac{1}{9}$ Three identical circles with radii 3 and centered at B,G,A respectively. The circumference of the circles pass through each other's centers. Find the area ...
1
vote
1answer
87 views

Vector VS Plane intersection

Could You help me with task: From point $M(3,5)$ that belongs to plane: $A(0,0), B(0,10), C(20,10), D(20,0)$, comes out vector $V$ at an angle a(with $OX$). Need to find point $X(x,y)$ at which he ...
2
votes
1answer
137 views

Chain of Circles

A chain of four circles centered at A, B, C, and D are touched on one side by the line GH and on the other side by a circular arc EF centered at O. Find the area of D in terms of the areas of A, B, ...
3
votes
2answers
290 views

How to derive this angle (about hyperbola)?

In section "Quadratic equation" of Hyperbola of Wikipedia( http://en.wikipedia.org/wiki/Hyperbola#Quadratic_equation), it it said that "The principal axes of the hyperbola make an angle Φ with the ...
0
votes
3answers
575 views

Find the area of the triangle using analytic geometry

We have a $\triangle ABC$ with: Base $AB$ with length 14 $AC$ with length 15 $BC$ with length 13 Find the area of the triangle using analytic geometry.
-1
votes
1answer
832 views

analytic geometry … 2 problems

1st problem : find the equation of the straight line having slope m passing through the point (a,0) . what are the coordinates of the point of intersection of this line with the y-axis ? 2nd problem : ...
2
votes
4answers
1k views

Equation for distance from a point outside a sphere to any point on its surface

I have a point m outside a sphere. The sphere center is o and r is the radius of sphere. Distance from point m to o is l. If we draw a line from m to any point on the surface of sphere, this line has ...
2
votes
2answers
1k views

Find the area of a triangle using analytic geometry

Given are the points $P (1,0)$ and $Q (3,2)$. The points $P$ and $Q$ have the same distance to a certain line $l$, which intersects the positive x-axis in the point $A$ and the positive y-axis in the ...
1
vote
2answers
855 views

Foci of Ellipse - given: Width and Height

Can you help me out with the next problem. I have an ellipse based on a width and a height. Is there any way you can find out where the focal points are? I need this information because I need to ...
2
votes
2answers
1k views

Proof that the Convex Hull of a finite set S is equal to all convex combinations of S

In $C^n$, how would you prove that the convex hull of a finite set $S$(convex hull being the intersection of all convex sets which contain $S$) is equal to the set consisting of all convex ...
0
votes
3answers
299 views

The equation of an ellipse

I have a couple of questions regarding ellipses. Get the equation of the ellips With Foci $(\pm 3,0)$ and which goes through $(2,\sqrt{2})$. This one I didn't understand AT ALL. I need some ...
1
vote
1answer
164 views

Equation of a parabola: Translations and directrixes

Find the equation of the paraboles, with: Focus $(3,0)$ and $x=-3$ is the directrix Focus $(0,2)$ and $y=-2$ is the directrix Vertex (I believe it is the vertex, the lowest/highest point) $(1,2)$ ...
2
votes
2answers
104 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
55 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
195 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
173 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
466 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
450 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? ...
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votes
2answers
1k 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
45 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, ...
2
votes
0answers
77 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 ...
0
votes
2answers
1k 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
301 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
389 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
933 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 ...
2
votes
4answers
12k 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
938 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
137 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
181 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
143 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
156 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
90 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
6k 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
328 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
96 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
178 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
173 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
49 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
266 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
98 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
515 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 ...