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

learn more… | top users | synonyms (1)

1
vote
1answer
36 views

difference between slopes of lines represented by an equation

The question is find the difference between slopes of lines give represented by equation of pair of lines which is $$x^2(\tan^2(\theta)+\cos^2(\theta))-2xy\tan(\theta)+y^2(\sin^2(\theta))=0$$ i have ...
-1
votes
1answer
43 views

prove that the quadrilateral $ABCD$ is a square

Given $ABCD$ a quadrilateral such that $AB\parallel CD$ and $\angle ACD=45^0, \angle A=90^0, \angle D=90^0 $ Need to prove that $ABCD$ is a square. I tried to use circles but it didn't help. Any ...
-4
votes
3answers
38 views

curve represented by the given equation. [on hold]

What does the equation $$x^2y-2xy-3y^2-4x^2+8x+12=0$$ represent. Im blank on it dont know how to proceed on it as all the equation seems to be quadratic in $x,y$
1
vote
3answers
48 views

The line $2x-y=5$ turns about a point…

The line $2x-y=5$ turns about a point on it, whose ordinate and abscissae are equal, through an angle of $45°$, in anti clockwise direction. Find the equation of line in the new position. My attempt ...
2
votes
3answers
34 views

How to find position of point that is x unit distant from AB line segment and y unit distant from BC line segment?

I am trying to calculate coordinates of point P, which is x units distant from AB line segment and y units distant from BC line segment. Edit: I am trying to write code for general solution. As ...
1
vote
1answer
18 views

Find the equation of the radius

Find the equation of the radius of the circle $x^2-4x-6y+y^2=23$ and passing through the point $(4,5)$. My attempt: Here the equation of the circle is: $$x^2-4x-6y+y^2=23$$ $$(x-2)^2+(y-3)^2=6^2$$ ...
1
vote
1answer
20 views

Intersection of cone and cylinder layout formula for sheet metal application

A common part in HVAC is a cylindrical pipe intersecting a truncated cone. I am designing a machine to mass produce this part. I would cut the parts out of sheet metal and roll them up to form the ...
1
vote
5answers
19 views

Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$.

Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$. Solution I calculated the two lines represented by ...
2
votes
2answers
37 views

If $ax^2+2hxy+by^2+2gx+2fy+c=0$, prove that

If $$ax^2+2hxy+by^2+2gx+2fy+c=0,$$ represents a pair of lines, show that the square of the distance from origin to their point of intersection is $$\frac{c(a+b)-f^2-g^2}{ab-h^2}.$$ My attempts; since ...
0
votes
1answer
24 views

Equation of pair of lines

Prove that the pair of lines $6x^2+5xy-4y^2+7x+13y-3=0$ form a parallelogram with the pair of lines $6x^2+5xy-4y^2=0$. Find its area. My attempt/ I factorized the second equation to get the two lines ...
0
votes
2answers
27 views

Equation To The Pair Of Angle Bisectors

Find the equation to the pair of angle bisectors of the pair of lines $(ax+by)^2=3(bx-ay)^2$. Efforts: $$(ax+by)^2=3(bx-ay)^2$$ After simplifying, I got: $$x^2(a^2-3b^2)+8abxy+y^2(b^2-3a^2)=0$$ Now, ...
0
votes
0answers
34 views

Are all curves with equation of the form $(\xi x +n) \cdot x = \text{const}$ circles?

Let $x(t)=(x_1(t),x_2(t))$ with $t\in [a,b]$ be a smooth curve in $\mathbb{R}^2$ and $\xi \in \mathbb{R}$ such that $$(\xi x +n) \cdot x = \text{const}$$ Here $n$ is the unit normal to the curve. Is ...
0
votes
0answers
21 views

locus of a variable straight line [closed]

Geometry: A variable straight line always intersects the lines x=c,y=0; y=c,z=0; z=c,x=0. find the equation to its locus. taking the equation of a line in parametric form and substitute the given ...
0
votes
0answers
20 views

Projection of a Vector on a Straight Line in $\mathbb{R}^3$

I have the following: Consider the straight line $(\epsilon)$ which passes through the origin and forms an angle $t$ with $Ox$ axis. Find the matrix $A$ which projects a random vector ...
0
votes
1answer
19 views

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem.

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem. I'm an high-speed aerodynamics student. I am studying a sweptback wing like in the figure below (in green). Notice that I ...
2
votes
1answer
31 views

Equation of plane perpendicular to given plane

Find the equation of the plane which contains the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ and which is perpendicular to the plane $5x+3y-6z+8=0$ By setting $z=0$ I found a ...
1
vote
1answer
20 views

Equation of line passing through origin

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\frac{\pi}{3}$ Now our required line should be ...
0
votes
0answers
54 views

Perimeter of a teardrop (made by two adjacent circles)

I'm trying to determine the perimeter of a teardrop shape formed by two adjacent circles (non-intersecting) with mutually tangent lines drawn on both sides of the circles. I've attached a sample ...
0
votes
1answer
57 views
+100

Trigonometric position function and intersection

I have the following position function for a point: $x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$ $y(t) := C_y - (S_x-C_x) \cdot ...
0
votes
3answers
33 views

Find the condition such that one of the lines defined by $ax^2+2hxy+by^2=0$ has slope $k$ times that of the other

Find the condition that the lines represented by $$ax^2+2hxy+by^2=0$$ are such that the slope of one line is $k$ times that of the other. I calculated the two represented by $ax^2+2hxy+by^2=0$ ...
3
votes
1answer
31 views

Expressing a point in $\mathbb{R}^n$ as a sum of unit vectors

I'm pretty sure that any point in $\mathbb{R}^n$ can be written as a sum of finitely many unit vectors (in $\mathbb{R}^n$, of course). However, I have no idea how to go about proving this. Any ideas? ...
0
votes
1answer
18 views

Proving the square formed by pairs of lines

Show that the two pairs of lines $12x^2+7xy-12y^2=0, 12x^2+7xy-12y^2-x+7y-1=0$ form a square. I know that both the equations represent a pair of straight lines. Also the first equation represents a ...
0
votes
1answer
21 views

Finding mid-point of $BC$ if point $A$, orthocenter and circumcenter are given in a triangle

If in a triangle $ABC$, $A \equiv (1,10)$, circumcenter $\equiv (-\frac13, \frac23)$ and orthocenter $\equiv (\frac{11}3, \frac43)$ then the coordinates of mid-point of side opposite to A is? ...
1
vote
1answer
21 views

Equation of a Pair of Straight Lines .2nd degree

Show that if one of the lines given by $a_1x^2+2h_1xy+b_1y^2=0$ coincides with one of the lines of $a_2x^2+2h_2xy+b_2y^2=0$ then $(a_1b_2 - a_2b_1)^2=4(a_2h_1 - a_1h_2)(b_1h_2-b_2h_1)$ Actually, I ...
0
votes
1answer
42 views

Proving equilateral triangle

Show that the lines $x^2+16xy-11y^2=0$ form an equilateral triangle with the line $2x+y+1=0$ and find its area. --------________________________--------- My solution is here; Here $x^2+16xy-11y^2=0$ ...
0
votes
1answer
28 views

No. of points determining a unique parabola

For a parabola, let Focus: $(a_1,b_1)$ Equation of directrix: $y-mx-c=0$ The equation of parabola is, $\sqrt{(x-a_1)^2+(y-b_1)^2}= \frac{|y-mx-c|}{\sqrt{1+m^2}}$ There are 4 parameters ...
0
votes
0answers
28 views

Finding the equation of diagonal

If $ax^2+2hxy+by^2=0$ be the two sides of a parallelogram and $px+qy=1$ is one diagonal then prove that the other diagonal is $y(bp-hq)=x(aq-hp)$. My solution is here; $ax^2+2hxy+by^2=0$ Multiplying ...
0
votes
5answers
74 views

Calculating the area of Triangle

Find the area of triangle formed by the lines $x^2+4xy+y^2=0$ and $x+y=1$. I know that the equation $x^2+4xy+y^2=0$ represents a pair of straight lines but how do i factorize it to get the two lines ...
1
vote
0answers
32 views

Equation of a plane given one point and two planes

I've done a question similar to this, however this one has no complete equations i can solve for. Determine the equation of the plane that passes through $(1,3,8)$ and is perpendicular to the line ...
0
votes
1answer
31 views

Find sides of a right triangle given hypotenuse c and area A (no numbers given)

I've solved couple of these, but I have no idea how to solve it without any numbers provided. I've tried using $A=\frac{ab}{2} \Rightarrow 2A=ab \Rightarrow 4A^2=a^2b^2$ and incorporating ...
0
votes
1answer
47 views

Question in the proof of the Brower fix point theorem

One can show that for any given homology theory $H$ with non-trivial coefficient group $G$ there does not exist a retract $\partial B^n \subset B^n$. Brower's fix point theorem states that any ...
0
votes
3answers
30 views

Are certain equations for orthogonal trajectories of a curve incomplete?

Suppose we wish to observe a Euclidean circle $C$ with radius $\alpha$. We define the relation $$R=\{(x,y)\in\mathbb{R}^2:\alpha^2=x^2+y^2,\text{fixed}\,\alpha\in\mathbb{R}^{+}\},$$ represented in ...
0
votes
0answers
18 views

How to assign variables to points on the cartesian plane?

Suppose I want to name the point $(3,4)$ a name say A. Can I just say $A=(3,4)$?
-1
votes
1answer
26 views

Equation of a circle whose radius and tangent is given

Equation of a circle which passes through the origin, whose radius is $a$ and for which $y = mx$ is a tangent.
0
votes
0answers
5 views

Why is convenient to consider points conjugate axis in hyperbola?

In "The Concise Oxford Dictionary of Mathematics" I found this: It may be convenient to consider the points $(0,-b),(0,b)$ on the conjugate axis, despite the fact that the hyperbola doesn't cut ...
2
votes
1answer
33 views

Rational points on a line

This question is quite unique. Does there exist some point in the coordinate system such that any line passing through it has at most 2 rational points lying on it?
0
votes
1answer
27 views

How can I find the location on the Z axis where two skew lines pass closest to each other on the XY plane?

I'm given one point $\overrightarrow{P}$ and the slopes $\frac{dX}{dZ}$ and $\frac{dY}{dZ}$ for each of my two lines. I'm trying to figure out where on the Z axis my lines pass closest to each ...
1
vote
1answer
31 views

Geodesic distance between equidistant points on a sphere [closed]

On the unit sphere equidistant points can be found for $1, 2, 3, 4, 6, 8, 12, 20$. The geodesic distance between the points are $\pi$ for $2$, $2\pi\over 3$ for $3$, $\pi\over 2$ for $6$, etc... Is ...
-1
votes
1answer
46 views

What is the equation of a pyramid with a square base?

Which algebraic description can be found for a pyramid, defined as a scalar function $$f:\mathbb{R}^2 \rightarrow \mathbb{R}$$ $$(x,y)\rightarrow z$$ Particular assumptions: Square base $z=0 \iff ...
0
votes
2answers
20 views

line parallel to x-axis and arbitrary intersection test

I am going through code snippets that calculate the x-intersection point between the line parallel to the x-ais and an arbitrary line between points (x1,y1) and (x2,y2). The code snippet does the ...
0
votes
1answer
31 views

General equation for a line contained in a plane and passing through a point

I have a vector $n$ and I seek a parametric equation for a line that is orthogonal to $n$ and passes through a point $(a,b,c)$. I got the equation of the plane formed by the normal vector and that ...
2
votes
1answer
34 views

Line equation through point, parallel to plane and intersecting line

Write the equations of the line that passes through point $M(1,0,7)$, is parallel with the plane $3x-y+2z-15=0$ and intersects line $\frac{x-1}{4}=\frac{y-3}{2}=\frac{z}{1}$ Alright, so from what I ...
0
votes
2answers
29 views

Height of a paralelogramm

I have the coordinates of the 4 vertexes of a parallelogram. If i calculate the length of two opposing sides, how do I get the perpendicular distance between them? Is it just the distance between the ...
0
votes
2answers
41 views

If a matrix $\boldsymbol{\mathrm {A}}$ is orthogonal, its determinant is $\pm 1$. Is the converse also true?

I know that an orthogonal matrix satisfies $$~~~~~~~~~~~~~~~~~~~~~~~~~\boldsymbol{\mathrm {AA}}^T=\boldsymbol{\mathrm {A}}^T\boldsymbol{\mathrm {A}}=\boldsymbol{\mathrm {I}}~~~~~~~~~~~~~~~~~~~~~~(*)$$ ...
3
votes
1answer
57 views

Triangle with same black and white areas

Suppose we have an infinite chessboard with the usual black/white coloring. A triangle $T$ with area $a$ is given with vertices at corners of some cells. Prove that there exists another triangle $T'$ ...
0
votes
1answer
28 views

Locus of vertex

A variable parabola of latus rectum $l $, touches a fixed equal parabola , the axes of the two curves being parallel . Then locus of the vertex of the moving curve is a parabola , then what is the ...
1
vote
2answers
48 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola [on hold]

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
2
votes
0answers
38 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
1
vote
2answers
48 views

Prob. 19, Chap. 1 in Baby Rudin: For what $\mathbf{c}$ and $r > 0$ does this equivalence hold?

Here's Prob. 19 in Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $\mathbf{a} \in \mathbb{R}^k$, $\mathbf{b} \in \mathbb{R}^k$. Find $\mathbf{c} \in ...
1
vote
0answers
32 views

Prob. 16, Chap. 1 in Baby Rudin

Here is Prob. 16, Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $k \geq 3$, $\mathbf{x}, \mathbf{y} \in \mathbb{R}^k$, $\vert \mathbf{x} - ...