1
vote
0answers
21 views

How to solve the sets of equations to find the matrix of coefficients for an ellipsoid

Regarding the below question: Finding equation of an ellipsoid I have two more questions: 1- In the "update" section of answer provided by @achillehui I can not understand the method he described for ...
2
votes
3answers
56 views

Finding the equation of a line whose segment is intercepted between axes

The question is: Find the equation of a line through (-2, 5) and whose segment intercepted between axes in the 2nd quadrant is 7√2 I have two graphs in mind but I don't know which one is correct. The ...
1
vote
2answers
122 views

Finding equation of an ellipsoid

Consider I have an ellipsoid (let say an egg) lies in a general form in 3D space. Suppose, I have the equations of two projected views of this egg (e.g. one projected view on x-y plane and another one ...
1
vote
0answers
39 views

Questions about circle

I found the following problem from a book. Let A = (-1, 0), B = (1, 0) and k = a constant which is not equal to 1. C(x, y) is a variable point such that AC = kBC. Find the locus of C. The ...
2
votes
3answers
47 views

Pair of straight lines

Question: Find the equation of the bisector of the obtuse angle between the lines $x - 2y + 4 = 0$ and $4x - 3y + 2 = 0$. I don't even know how to proceed here. I know how to find the angle between ...
1
vote
2answers
22 views

How to define a cloud of points relative to a vector path?

I've been researching and playing with examples of particle clouds in a graphics visualization. Most use shape geometries to define a field of particles, or parameters for distributing them randomly ...
1
vote
0answers
20 views

Equation of a line through a point and another line

I need to get the equation of a line that passes through the point Q(6, 3, 2) and intersects: $$L: (1, -1, 4) + t(0, -1, 1)$$ and forms an angle of 60° What I did so far: The direction vector of L ...
3
votes
2answers
25 views

Parallel plane that contain lines

I have the lines: $$L_1: \frac{x-3}{2}= \frac{y+5}{-3} = \frac{z+1}{5} ,$$ $$L_2: \frac{x+1}{-4}=\frac{y-1}{3}=\frac{z-3}{-1}$$ I need the equations of the parallel planes $P_1$ and $P_2$ that ...
2
votes
3answers
107 views

What is the cone of the conic section?

Given the general (real valued) equation of a conic section: $$ A x^2 + B xy + C y^2 + D x + E y + F = 0 $$ Then what is the circular cone associated with it ? Is it unique ? And is there a way to ...
1
vote
2answers
39 views

Rotation around a line which is determined by two points in 3D space

If we have three points like $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$ and $C(a,b,c)$. Then, $A$ and $B$ determines a line like $l$. After that, we rotate $C$ around $l$ by $\omega$ degree (anti-clockwise). ...
0
votes
0answers
35 views

A book on analytic geometry

It's easy to find good recommendation for books here for any subject other than analytic geometry ,therefore I'd like to ask for any suggestion of analytic geometry books ,the only charactrestic I'm ...
1
vote
1answer
35 views

Lines joining origin to points of intersection of two conics

If the lines joining origin and point of intersection of curves $$ax^2+2hxy+by^2+2gx=0$$ and $$a_1x^2+2h_1xy+b_1y^2+2g_1x=0$$ are mutually perpendicular, then prove that $$g(a_1+b_1)=g_1(a+b)$$ How ...
2
votes
1answer
53 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
1
vote
3answers
37 views

Doubts on locus and its equation

"Find the equation to the locus of a point which is collinear with points $M(a,0)$ and $N(0,b)$." The answer is $- x/a + y/b$. How I tried to find the solution: $P$ is a point whose assigned ...
0
votes
1answer
31 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
-1
votes
3answers
115 views

getting the slopes of the sides of an equilateral triangle given 2 points

I want to get the slopes of an equilateral triangle given the 2 vertices. Let's say they are (0, 0) and (5, 5). Graphing this would give 2 triangles forming a diamond. I tried to use distance formula ...
5
votes
3answers
209 views

Calculate the distance from a point to a line

Por favor, alguém me ajude com essa questão de Geometria: Please, can someone help me with this geometry question? Given the point $A(3,4,-2)$ and the line $$r:\left\{\begin{array}{l} x = 1 + t ...
0
votes
1answer
39 views

Central angle of an ellipse

If I have an ellipse centered at the origin and know the length of $a$ and $b$ and was given the length of an arc, how can I find the angle that is between the two radius from the center of the ...
-1
votes
1answer
19 views

Preserving incidence relation proof

How can one prove via analytic method that projective map preserves incidence relation?
0
votes
1answer
19 views

Determine Center Point based on 2 separate elipses

First timer here. I've been digging back into my good old maths days but am extremely rusty (beyond belief). I got a really tricky question that i want to determine formula for so that my mate can ...
-1
votes
4answers
53 views

Find the line through $(-1,4)$ for which the distance to $(6,3)$ is 5

This is the question: Find the line through $(-1,4)$ for which the distance to $(6,3)$ is $5$ The answer is: $y-4=-4/3(x+1)$ and $y-4=3/4(x+1)$ I do not know how to get this answer. ...
2
votes
1answer
28 views

Definition of (hyper)planes

I know the definition of a plane to be: $(r-r_0)\cdot n = 0$ where $n$ is the vector perpendicular to the plane, $r$ the vector to a given point and $r_0$ the vectors to the points which constitute ...
2
votes
3answers
44 views

Find a specific vector equation of a line that divides a angle in half.

I've been studying a little geometry on my own, and I just recently stumbled on this problem, that I'm unable to answer: Given the points A=(2,-1), B=(5,4) and C=(-7,8), find a vector equation of a ...
1
vote
1answer
33 views

Doubt on locus of a median point

I'm learning about geometric locus and ain't had an good time, I'm struggling with this problem: By the way, any study resource on geometric locus is welcome! Given an segment $AB$ formed by points ...
1
vote
2answers
56 views

Locus of vertex of triangle moving inside circle

A right triangle with sides $3,4$ and $5$ lies inside the circle $2x^2+2y^2=25$. The triangle is moved inside the circle in such a way that its hypotenuse always forms a chord of the circle. The locus ...
1
vote
1answer
101 views

Analytic geometry - rotation + translation

In $K=O\vec{e_1}\vec{e_2}\vec{e_3}$ I have to find the analytical representation of the screw motion( rotation + translation) $\psi$ with a rotation axis $g$ given by the points $A(5,-4,3)$ and ...
0
votes
0answers
44 views

Center of circumscribed circle of a triangle

I've been given the parametric equations of the height, median and inner angle bisector through the point $A$ of a triangle $\triangle{ABC}$: $$ h: \begin{cases}x = 2 - s \\ y = 1 \\ z = -3 + 2s ...
0
votes
2answers
43 views

Closest Point on a Sphere to Another Point

Given a sphere $S(c,r)$, $c$ being the center point $(x,y,z)$ and $r$ being the radius, there is a point $p(x', y', z')$ which is either inside or outside $S$. I want to find the point $q$ such that ...
2
votes
1answer
32 views

$\frac{|| \overline{AM}||}{|| \overline{AB}||}=\frac{|| \overline{AN}||}{|| \overline{AC}||}=\frac{|| \overline{MN}||}{|| \overline{BC}||}$

$\Delta ABC$ is a triangle, $M$ is a point in the segment $\overrightarrow{AB}$ and $N$ is a point in the segment $\overrightarrow{AC}$, such that $\overrightarrow{MN}$ is parallel to ...
0
votes
4answers
503 views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
votes
2answers
21 views

Finding the equations of surfaces of revolution

I have the following question: $$\text{Sketch and find the equations of the surfaces formed by}$$ $$\text{i) }x^2 - y^2 + 1 = 0 \text{ about the y-axis}$$ $$\text{ii) }x^2 - 2y^2 + 2a^2 = 0 \text{ ...
1
vote
1answer
51 views

Calculating the adjustment translation to be applied after rotating and scaling so that operations pivot about a given point.

I have a matrix for transforming an image into a target frame. The matrix is a function of a scale, $s$ rotation angle, $\theta$, and a translation that is applied after rotating, $tx, ty$. The ...
1
vote
2answers
55 views

Solving a geometric question without trigonometric tools.

$AB$ is a diameter in a circle. from point $C$ outside the circle passing $2$ intersects to the circle at points $A$ and $B$. $AC$ cuts the circle at point $F$ and $BC$ cuts the circle at point ...
1
vote
3answers
95 views

Moments at which moving points on a circle coincide

Points A $(0,1)$ and B $(1,0)$ start moving along the circumference of a unit circle with center $(0,0)$ in the same, positive (that is, counterclockwise) direction. Every minute, points A and B ...
0
votes
2answers
43 views

Length of a line in an isosceles triangle. (mind boggling )

In an isosceles triangle ABC, side AB and AC are equal in length. There exists a point D on the side AB. The angle BAC is theeta . The side AD is two units smaller than AC .What is the generalized ...
26
votes
3answers
1k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room, which is embodied in the image below. Notice how the light is being cast on the wall? There is a curve that defines the ...
0
votes
1answer
49 views

How to find the midpoint given the linear equation

Given the linear equation: $$y = 7247.5188 -2395.0376x$$ how do I find the midpoint of this line?
1
vote
0answers
43 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
1
vote
1answer
44 views

Calculating tangent on ellipse

I want to calculate the slope of the tangent at one point of an ellipse whose centre is shifted towards the coordinates $(x_c;y_c)$ and also rotated by an angle $\alpha$ around its centre. Now, I have ...
1
vote
4answers
78 views

A triangle has to find its third side.

Problem: (Euclid had a triangle in mind - I am including this line so that future googles come across this question) The triangles longest side is $20$ and another side is $10$. Its area is $80$. ...
1
vote
1answer
39 views

Crazy rectangles, semi-circles, and circles!

Problem is to find the ratio of the area of the circle to that of the semi-circle. Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right ...
0
votes
1answer
18 views

Proving congruency of triangles

Question: Given $AB$ is diameter, $C$ and $D$ lie on circumference, $AB = 15cm$, $AC = 12cm$, $BD = 9cm$, find area of quadrilateral ABCD. Note that the points $O$ and $Q$ were not in the ...
0
votes
1answer
19 views

How to prove that given a line L prove that all points of a fixed distance k form two lines parallel to L

How can I prove start this?I know intuitively since they never meet they are parallel, but I don't think that is a direct proof.
1
vote
1answer
31 views

Geometric Intuition Of Partial Derivative in 2D space

$$f(x,y)=3x^2+5y^2-2x+3y+7=0\\f'x=6x-2=0\\x=\frac {1}{3}\\f'y=10y+3=0\\ y=\frac {-3}{10} \\ O(\frac {1}{3},\frac {-3}{10})$$ As you see, we've used the partial derivative to find the center point of ...
2
votes
2answers
31 views

Prove $(|OP|)+ |PQ|)^2 > |OQ|^2$

I did all the algebra and for some reason I'm getting 0 > $y_2^2$ which is clearly wrong. Where did I mess up at?
2
votes
1answer
30 views

Given two closed curves, when is their minkowski sum differentiable?

Suppose you are given closed curves, $\gamma_1$ and $\gamma_2$, which define convex figures in the plane. If we take the minkowski sum of $\gamma_1$ and $\gamma_2$, when is the resulting curve ...
0
votes
1answer
45 views

Geometric interpretation of a complex solution

A straight line in 2-D $x+y=3$ and a circle in 2-D $x^2+y^2=4$ do not have a point of intersection in the plane containing the two. But on solving these equations analytically, on gets 2 complex ...
1
vote
2answers
71 views

Maximize the distance between a point and a bounding rectangle

There are $n$ random points in the $x-y$ plane, whose coordinates are known beforehand. We can use a minimum bounding rectangle (MBR) to bound these points. In this scenario, the MBR can be rotated, ...
1
vote
1answer
30 views

need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
2
votes
2answers
64 views

Orthogonal tangents to an ellipse [duplicate]

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...