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

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41 views

Equation to the circle.

How to show that the equation to the circle of which the points $(x_1,y_1)$ and $(x_2,y_2)$ are the ends of a cord of a segment containing an angle $\theta$ is, $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2) ± ...
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1answer
11 views

How to derive the volume of a tetrahedron with the following data? [on hold]

The vertices of a tetrahedron are:- A - (0, 0, 0) B - (0, 0, a) C - (0, b, 0) D - (c, 0, 0) Prove that the volume is:- 1/6 abc. A figure will be helpful.
4
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3answers
64 views

Are planes in $3$-dimensions two-dimensional?

Are planes in $3$-dimensions two-dimensional? The reason I ask is because mathematically the $xy$-plane exists in $3$D space but appears to be $2$D, but how can something $2$D be in $3$D space? I ...
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0answers
15 views

If two monic polynomials have no common roots, are the coefficients of their product locally diffeomorphic to the product of the coefficients?

Let $P^d (t,\lambda)$ be the "generic" d-th degree monic polynomial $P^d (t,\lambda) = t^d + \sum\limits_{i=1}^d \lambda_i t^{d-i}$ with real coefficients. Let $\lambda(\xi,\eta)$ be given by the ...
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2answers
21 views

Find the equation of the line which is

Find the equation of the line perpendicular to the line joining the points $A(3,6)$ and $B(-6,9)$, which divides the line $AB$ in the ratio of $2:1$. My attempt: Equation of $AB$ is ...
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1answer
24 views

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$ [on hold]

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$. Could some explain how to solve this in detailed.
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1answer
27 views

angle between given pair of lines. [on hold]

If the angle between the lines represented by $$2x^2+5xy+3y^2+6x+7y+4=0$$ is $\tan^{-1}(m)$ , and $a^2+b^2-ab-a-b+1\leq 0$ then $2a+3b=?$ ATTEMPT now what i found is the value of tan from given "pair" ...
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1answer
48 views

How do I find the difference between the gradients of two lines represented by an equation

I want to find the difference between the gradients (or slopes?) of two lines. The equation of the lines is $$x^2(\tan^2 \theta+\cos^2 \theta)-2xy\tan\theta+y^2 \sin^2 \theta=0$$ I have assumed the ...
-1
votes
1answer
56 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 ...
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3answers
50 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 $x,y$
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3answers
51 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
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3answers
37 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
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1answer
20 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
23 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 ...
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5answers
20 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
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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
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1answer
32 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 ...
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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 ...
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0answers
55 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 ...
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1answer
67 views

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 ...
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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
32 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
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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
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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? ...
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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 ...
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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
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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 ...
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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 ...
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5answers
84 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $A=(3,4)$?
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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.
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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
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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?
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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 ...
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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 ...
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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
32 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 ...