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

learn more… | top users | synonyms (1)

1
vote
1answer
85 views

Canonical equation of a line in space: horizontal and vertical lines

I have a question about canonical equation of a line in 3d space: how can I handle vertical and horizontal lines? One of direction vector's values will be just $0$, but this will mess up the equation, ...
1
vote
0answers
18 views

Polynomial = 0 on edges of tesseract?

Can there exist a polynomial on $w,x,y,z$ whose value is zero on the edges of a tesseract – or rather, on the projection of those edges to the unit sphere – and nonzero everywhere else on the unit ...
0
votes
1answer
116 views

Equation of line of shortest distance

Given two lines $r_1=3i+5j+7k+m(i-2j+7k)$ and $r_2=-i-j-k+n(7i-6k+j)$ how to find the equation of line of shortest distance? I found the direction ratios of line of shortest distance as (2,3,4) but ...
0
votes
0answers
57 views

Perpendicular distance between point and plane

I have been working on fitting a plane to 3d points and then calculating the perpendicular distance between each point and the plane using Matlab. So far I can find the plane equation in $Ax+By+Cz+D = ...
0
votes
1answer
16 views

Finding the symmetric line of another line with regard to a plane

I've found this challenge to solve that I have no idea where to start. We are supposed to find the symmetric line to this one $r: (10,1,2) + \lambda (3,1,1)$ with regard to the plane $\alpha \ \{ ...
0
votes
3answers
49 views

Finding the Angle Using the Unit circle?

Find the angle between the planes $ 5(x − 1) − 3(y + 2) + 2z = 0,$ $ x + 3(y − 1) + 2(z + 4) = 0.$ I am not too sure how to start and solve this problem I know the formula $\cos \theta=|U_{n_1}*U_{...
1
vote
1answer
33 views

Converting coordinates Points to Slope Intercept Form?

Write the equation of the plane in intercept form and find the points where it intersects the coordinate axes. $4x + 5y − 6z = 60.$ Is there a way to algebraically using y = mx+b to convert this to ...
1
vote
3answers
101 views

Prove that $-3/2\leq\cos a + \cos b + \cos c\leq 3$?

Given 3 non-null vectors $v,u,w$ and angles $a=(u,v), b=(u,w), c=(v,w)$, Prove that $-3/2\leq\cos a + \cos b + \cos c\leq 3$. I've managed to prove that: $\cos a + \cos b + \cos c\leq 3$ basically ...
1
vote
2answers
55 views

What is the largest area from this following triangle?

Given coordinates A(2,0), B(0,2), and C is in line x+y= 5. We want to form the triangle ABC from these coordinates. What is the largest possible area for this triangle? My thinking about this ...
0
votes
1answer
14 views

Prove that the biscetors of adjacent suplementary angles are perpendicular?

I did the following: Take the vectors $A=(a,0), B=(x,y), -A=(-a,0)$. Then multiply each vector $X$ by $\cfrac{1}{|X|}$ to obtain the unit vector. (I don't know if this step is really necessary.) ...
1
vote
1answer
36 views

Finding out which points lie on this equation?

Which of the points P(3, 2, 1), Q(2, 3, 1), R(1, 4, 1) lie on the plane $3(x − 1) + 4y − 5(z + 2) = 0?$ The equation I know is $3x + 4y + 5z.$ do I just graph this or do I plug in the points to find ...
3
votes
2answers
41 views

Finding Points That Lie On A 3D Line?

Which of the points $P(1, 2, 0), Q(−5, 1, 5), R(−4, 2, 5)$ lie on the line $l : r(t) = (i + 2j) + t(6i + j − 5k)?$ $l3 : r3(ν) = (6i − j) − ν(2i − 4j + 6k),$ $l4 : r4(w) = (1/2+ 1/2w)i − wj − (1 + 2/...
6
votes
2answers
105 views

Find the area of the circle that falls between the circle $x^2+y^2=5$ and the lines $x^2-4y^2+6x+9=0$

Find the area of the circle that falls between the circle $x^2+y^2=5$ and the lines $x^2-4y^2+6x+9=0$. I tried to solve this question. The lines are $x-2y+3=0$ and $x+2y+3=0$ which intersect at $(-3,...
6
votes
1answer
81 views

Deducing formulas of analytic geometry

The way I learned analytic geometry in highschool, I was given a lot of formulas and shown what they do and maybe, just maybe, why they are accurate. So, an ellipse was defined as being a curve that ...
0
votes
1answer
38 views

Hint for show that $\sin AOB \sin COD + \sin AOC \sin DOB + \sin AOD \sin BOC = 0$

I need a hint for proof of: If $OA, OB, OC, OD$ any four lines meeting in a point $O$, show that $$\sin AOB \sin COD + \sin AOC \sin DOB + \sin AOD \sin BOC = 0$$ I'm thinking of applying several ...
0
votes
0answers
19 views

General concepts of planes

My textbook has a question that we are supposed to answer true or false for general plane and line concepts. The sentences are: If two planes are perpendicular to a same line, then they are ...
2
votes
1answer
24 views

General concepts of lines and planes

I've come across an exercise that asks the reader to state the correct answer. For me, the only one right is the third, but I would like to know why the first one is not correct also: The line $r$ ...
1
vote
2answers
60 views

Rotation in multidimensional space

I have a regular pentagon, with coordinates \begin{align*} x(1) &= (2, 0, 1, 1, 0) \\ x(2) &= (1, 1, 0, 2, 0) \\ x(3) &= (0, 2, 0, 1, 1) \\ x(4) &= (0, 1, 1, 0, 2) \\ ...
1
vote
1answer
29 views

Line through two points in Euclidian space

We have two points $a,b \in \mathbb{R}^n$. The line connecting $ a$ and $b$ equals $$ \{ \lambda a + \mu b : \lambda, \mu \ \in \mathbb{R} \}$$ with $\lambda + \mu = 1$. I want to prove this claim ...
5
votes
2answers
71 views

Intersection of hyperplanes

We have two vectors, $x = (1,4, -1)$ and $y = (-1,0,1)$ in $\mathbb{R}^3$. We have the following hyperplanes: $P_1 = \{ v \in \mathbb{R}^3 : \langle x, v \rangle =2\} $ $P_2 = \{ v \in \...
4
votes
0answers
155 views

“Hard” exercises on Linear Algebra and Analytic Geometry

I started lecturing this subject called "Linear Algebra and Analytic Geometry" and in the second day of class I was approached by an undergrad student, asking for referenced that would contain "hard" ...
2
votes
2answers
221 views

Lines and planes - general concepts

I've come across a book that has this general questions about lines and planes. I can't agree with some of the answers it presents, for the reasons that I'll state below: True or False: Three ...
0
votes
1answer
29 views

Prove that the envelope of the family of lines $(\cos\theta+\sin\theta)x+(\cos\theta-\sin\theta)y+2\sin\theta-\cos\theta-4=0$

Prove that the envelope of the family of lines $(\cos\theta+\sin\theta)x+(\cos\theta-\sin\theta)y+2\sin\theta-\cos\theta-4=0$ I did not know much about how to find envelope of a curve.I read on ...
0
votes
1answer
152 views

Cartesian coordinates for vertices of a regular polygon?

I'm trying to draw: A set of $N$ (edit) irregular polygons one inside the other, where the innermost should be an equilateral triangle, enclosed by a square, enclosed by a pentagon, etc. Where ...
0
votes
2answers
121 views

Equilateral Triangle Property

If the vertices of a triangle have integral coordinates how to prove that the triangle cannot be equilateral ?
2
votes
2answers
59 views

Finding A Quadratic Whose Roots Equals Intercept On Axes and Area Equals A

How to find the quadratic equation whose roots are the x and y intercepts of the line passing through $(1,1)$ and making a triangle of area A with the axes? Ok I'm getting $(1-m)(1-1/m)=A$ and $(x-(...
0
votes
0answers
15 views

Given the tetrahedron $OABC$, find a condition on $a OA+ b OB + c OC$ such that this is always inside $ABC$.

I did the following: Taking the tetrahedron $OABC$, one can decompose it in: $OA,OB,OC, AB,BC$. And then, writing: $$x(BC-AB)+AB\quad x\in[0,1]$$ We obtain all the points in the line segment from $...
1
vote
2answers
55 views

General equation of line that goes through center of a circle and a point

Given an arbitrary point $P$, at $(x_{1}, y_{1})$, is there a general expression of a line that goes through a circle of radius $r$ centered at the origin? I know there are infinite number of such ...
3
votes
1answer
44 views

In the triangle $ABC$, if $a=\frac{2(b^2-c^2)}{-b+\sqrt{b^2+4c^2}}$, prove that $3\cdot\widehat{C}=2\cdot\widehat{B}$.

Just like in the title, I have to prove that if in a triangle $ABC$ $$a=\frac{2(b^2-c^2)}{-b+\sqrt{b^2+4c^2}}$$ holds, then $3\cdot\widehat{C}=2\cdot\widehat{B}$. The denominator of the big ...
-1
votes
2answers
42 views

Area formed by line and circle w.r.t origin

A straight line is drawn through the center of the circle $x^2+y^2-2ax=0$, parallel to the straight line $x+2y=0$ and intersecting the circle at $A$ and $B$. Then area of triangle $AOB$ is? What is ...
1
vote
1answer
34 views

What is the shortest method to solve this sum?-Pair Of Straight Lines

What is the shortest method to solve this sum? One of the bisector of the angle between the lines $a(x-1)^2+2h(x-1)(y-2)+b(y-2)^2=0$ is $x+2y=5$.The other bisector is what? My approach is ...
0
votes
1answer
30 views

Write $CX,AY,BZ$ in terms of $CA,CB$ and the ratios $\alpha, \beta, \gamma$?

The point $X$ divides $AB$ in the ratio $\alpha$, $Y$ divides $BC$ in the ratio $\beta$ and $Z$ divides $CA$ in the ratio $\gamma$. Write $CX,AY,BZ$ in terms of $CA,CB,\alpha, \beta, \gamma$. I did ...
2
votes
1answer
92 views

Show that the locus of the centroids of equilateral triangles inscribed in the parabola $y^2=4ax$ is the parabola $9y^2-4ax+32a^2=0.$

Show that the locus of the centroids of equilateral triangles inscribed in the parabola $y^2=4ax$ is the parabola $9y^2-4ax+32a^2=0.$ I tried to solve it.I took three coordinates of the equilateral ...
2
votes
1answer
131 views

Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection.

Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection. I tried but could not reach final answer.The tangent ...
3
votes
1answer
239 views

Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$

Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$ The required condition is $a^2>8b^2$.I dont know how ...
1
vote
1answer
40 views

Analytically Understanding The Definite Integral As A Limit Of Sums

With naive intuition one can obviously see that the definite integral as infinite subdivisions of an area under a curve, within the finite interval "a to b", from which the function of integration ...
1
vote
1answer
67 views

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola.

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola. I tried to solve it but failed.Can someone please ...
2
votes
0answers
25 views

Locus of point satisfying a condition

Consider a fixed point $O$ and $n$ fixed straight lines. Through $O$ a variable line is drawn intersecting the fixed lines in $P_1,P_2,\ldots,P_n$. On this variable line, a point $P$ is taken such ...
0
votes
1answer
25 views

Coordinate Geometry:Locus Based Problem

A rod AB of length l slides with its ends on the coordinate axes.Let O be the origin.The rectangle OAPB is completed. How to prove the locus of the foot of perpendicular drawn from P onto AB is $x^...
4
votes
1answer
256 views

Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$.

Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$. I have encountered this question from SL Loney.I have ...
0
votes
0answers
32 views

Straight Lines and Cordinate Geometry

The Locus of the point $P$, such that sum of squares of its distances from $(1,2)$ and $(3,4)$ is $25$ units, is $x^2+y^2-4x-6y+k=0$. Then $k =$ ?
1
vote
1answer
119 views

Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...
1
vote
1answer
35 views

Expression of reflection isometry in the complex plane

Using the fact that an anti-displacement in the plan has the form $$f(z) = a \overline{z} + b$$ I have done some computation to find the reflection about the line passing through two points $P$ and ...
2
votes
0answers
67 views

What theorems or frameworks explain why the roots of well-behaved functions $h : \mathbb{R} \leftarrow \mathbb{R}^2$ seem to be made up of “pieces”?

First, some terminology: given functions $g,f:Y \leftarrow X$, the equalizer of $g$ and $f$ is defined to be the set of all solutions $x \in X$ to the equation $g(x)=f(x)$ in $Y$. Okay. The following ...
7
votes
2answers
97 views

How are asymptotes actually defined in rigorous mathematics?

This question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x \...
1
vote
0answers
17 views

Request reflection matrix about these types

Supposed there's $(a,b)$ point and going to be reflected and find the mapping. The baseline formula will I use is $\begin{pmatrix} x' \\ y' \end{pmatrix}=M_{R} \begin{pmatrix} x \\ y \end{pmatrix}$. ...
0
votes
2answers
48 views

Single transformation matrix of $A \circ B$ and $B \circ A$ with certain conditions

Let $A$ is 2x1 translation matrix and $B$ is 2x2 matrix of reflection or rotation matrix (reflection, rotation, etc.). Suppose I want to find the mapping of a $y=mx+c$ line and the mapping is done by ...
0
votes
1answer
43 views

Rotation and translation of coordinate axes

I am studying rotation and translation of conical but have no doubt in basic concept (Sorry, I know this is a very stupid question but I'm really struggling to understand). Especially in this equation:...
2
votes
0answers
58 views

Eccentricity of $9x^2 + 4y^2 - 24y + 144 = 0$

For a National Board Exam Review: Compute the eccentricity of a given curve $9x^2 + 4y^2 - 24y + 144 = 0$ Answer is $0.75$ I try: $$9x^2 + 4y^2 - 24y + 144 = 0$$ $$9x^2 + 4(y^2 - 6y + 9) = -...
0
votes
1answer
341 views

Finding the Equation of an Ellipse given the Length of the Latus Rectum and the Distance between the Foci

For a National Board Exam Review: Find the equation of the ellipse having a length of latus rectum of ${ \frac{3}{2} }$ and the distance between the foci is ${ 2\sqrt{13} }$ Answer is ${ \frac{x^...