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

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4
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1answer
69 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
29 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
15 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
65 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
0answers
14 views

Curvature from 3 Points

I have 3 GPS points (latitudes & longitudes in spherical co-ordinates) and I need to calculate the curvature. How can this be done?
0
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2answers
44 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
15 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 ...
0
votes
0answers
9 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 ...
0
votes
2answers
23 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
26 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
36 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 ...
0
votes
1answer
19 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 ...
-1
votes
0answers
40 views

Three line segments made by intersection in harmonic progression [on hold]

I'm learning coordinate geometry in high school and have this question as a doubt. The equations of three lines are $7x + y = 16$ , $5x - y - 8 = 0$ and $x - 5y + 8 = 0$. A variable line through ...
0
votes
1answer
21 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 ...
2
votes
1answer
29 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
58 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 ...
0
votes
0answers
30 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
33 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 ...
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vote
0answers
42 views
+50

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.

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. I tried to solve it but failed.Can someone please help me to ...
2
votes
0answers
24 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
9 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 ...
3
votes
1answer
59 views
+50

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 ...
-2
votes
1answer
15 views

Intersection point [closed]

I have line coordinate points and circle centre coordinate points and radius of the circle. I want to find the intersection point of circle and line using these coordinates and circle radius
-1
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0answers
38 views

Equilateral triangle [closed]

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are $(0,\,4)$ and $(0,\,0)$, find the third vertex. How many triangles are ...
-2
votes
0answers
8 views

Coordinate Geometry and Straight Lines

Let A and A' be points (5,0) and (-5,0) respectively. The equation of the locus of all points P(x,y) such that ||AP|-|A'P||=8 is ?
0
votes
0answers
26 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
40 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 ...
0
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0answers
21 views

Compute outer and inner outlines of graph of curves

Let's have some cubic Bezier curves and straight lines. Some of the Bezier curves and straight lines might have a shared start or end point, some might intersect. Input: A list of cubic Bezier ...
1
vote
1answer
26 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
59 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
69 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
8 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
35 views
+50

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
18 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 ...
2
votes
0answers
42 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
0answers
19 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 ${ ...
1
vote
1answer
13 views

Converting a plane from Cartesian to Parametric

Find the equations of the following plane in both cartesian and parametric form: The plane through the point $(1,4,5)$ and perpendicular to the vector $(7,1,4)$. So far, I have obtained the ...
-2
votes
1answer
15 views

Rotate vectors around an arbitrary point on a plane

Is it possible to rotate vectors around a point on a plane? say I have some plane ax+by+cz+d=0 and a vector in form of (x0, y0, z0)+ (i, j,k)t The rotation point is in form of (x1, y1, z1) how do ...
1
vote
1answer
22 views

Translate 2x^2 -8xy+4x+12 into the Standard form of a Hyperbola; Second Degree Term Missing

For a National Board Exam Review: What conic section is ${ 2x^2 -8xy+4x+12 }$ ? Answer is Hyperbola. But I can't seem to translate it properly to the standard form of a hyperbola.. What am I ...
0
votes
2answers
19 views

Finding the Width at the Bottom of a Vertical Parabolic Arc

For a National Board Exam Review: An arc 18m high has the form of a parabola with the axis vertical. If the width of the arc 8m from the top is 64m, Find the width of the arc at the bottom. ...
0
votes
0answers
25 views

Conic classification

I have a formula any and wonder what that is equation (hyperbola, point, lines, ellipse, parabola etc.) . However, I have doubts when I do the translation and rotation of coordinate systems. I know ...
1
vote
2answers
47 views

What is the general equation equation for rotated ellipsoid?

I have general equation for ellipsoid not in center: $$ \frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}+\frac{(z-z_0)^2}{c^2}=1.$$ What is the equation when it's rotated based on $\alpha$(over $x$ axis), ...
0
votes
0answers
28 views

Why is a positive definite matrix needed in the ellipsoid matrix representation?

An ellipsoid centered at the origin is defined by the solutions $\mathbf{x}$ to the equation $\mathbf{x}^TM\mathbf{x} = 1$, where M is a positive definite matrix. How can I see why M needs to be ...
1
vote
2answers
31 views

$\left(a-\frac{1}{r^2}\right)\left(b-\frac{1}{r^2}\right)=h^2$

If $ax^2+2h xy+by^2=1$,prove that the maximum and minimum values of $x^2+y^2$ are given by the values of $r^2$ satisfying the relation $\left(a-\frac{1}{r^2}\right)\left(b-\frac{1}{r^2}\right)=h^2$ ...
1
vote
3answers
34 views

How to Translate two Equations for a “+/-”

For a National Board Exam Review: Find the Equation for the Asymptotes of a Hyperbola ${ (y-x)^2 - (x+5)^2 = 36 }$ Answer is ${ y-5 = \pm (x+5) }$ I've already solved the equations: here they ...
2
votes
1answer
27 views

Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle

Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle.The centroid of the equilateral triangle is ...
1
vote
1answer
29 views

Prove that distance of $P$ from either of the points of contact is $\sqrt{\frac{abc}{a+b+c}}$

Three circles of radii $a,b,c$ touch one another externally and the tangents at their points of contact meet at a point $P$.Prove that distance of $P$ from either of the points of contact is ...
72
votes
9answers
3k views

Prove the theorem on analytic geometry in the picture.

I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in ...
-1
votes
0answers
33 views

Plot Points Along plane at a distance

I have a plane $0.176776695x+0y−0.176776695z+0.35355339=0$ I select an arbitrary point $P$ on the plane. $P =(1,2,3)$ Now I want to find another point "$Q$" on the same plane that is offset $0.25$ ...
1
vote
2answers
49 views

Determine the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ such that it has the least area but contains the circle $(x-1)^2+y^2=1$

Determine the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ such that it has the least area but contains the circle $(x-1)^2+y^2=1$ Since the area of ellipse is $A=\pi ab\Rightarrow ...