Tagged Questions

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

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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, ...
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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 ...
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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 ...
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“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" ...
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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 ...
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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 ...
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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 ...
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Equilateral Triangle Property

If the vertices of a triangle have integral coordinates how to prove that the triangle cannot be equilateral ?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\begin{pmatrix} x' \\ y' \end{pmatrix}=M_{R} \begin{pmatrix} x \\ y \end{pmatrix}$. ...
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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 ...
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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:...
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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) = -...
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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^...