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

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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 ...
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1answer
15 views

Intersection point [on hold]

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
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36 views

Equilateral triangle [on hold]

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 ...
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1answer
22 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 ...
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2answers
65 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 ...
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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 ?
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1answer
295 views

To find an intersection point between two planes with only the direction vector

Find the intersection between two planes $x−3y−2z = 2$ and $2x+y+3z = 1$ Solution: $(1)$$\quad n_1 \times n_2 =\langle −7,7,7\rangle =7 \langle −1,1,1\rangle$. $(2)$ To find one intersection ...
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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 $k =$ ?
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1answer
23 views

What is the domain of the given function with the greatest integer?

The domain of the function $$f(x)=\sqrt{\frac{4-x^2}{[x]+2}}$$ where $[x]$ represents the greatest integer function, is (a) $(-\infty,-1)\cup[-1,2]$ (b) ...
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1answer
33 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 ...
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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 ...
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19 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 ...
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0answers
52 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 ...
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1answer
122 views

Analytical Geometry problem with complex numbers - alternate solutions.

The question is to show that the equation of the lines making angles $45^\circ$ with the line: $$ \bar{a}z + a\bar{z} + b = 0; \;\;\;\;\; a,z \in \mathbb{C}, b \in \mathbb{R} $$ and passing through a ...
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2answers
27 views

Intercept made by a line between two concentric circles

Let $$x^2+y^2-9=4r^2\enspace (r=1,2,3)$$ be $3$ concentric cirlces. Prove that the intercept made by line $$3x+4y+15=0$$ between any two cirlces is same. I thought of calculating the intercept ...
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5answers
18k views

What is the general equation of the ellipse that is not in the origin and rotated by an angle?

I have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?
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2answers
22 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 ...
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0answers
7 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 $\begin{pmatrix} x' \\ y' \end{pmatrix}=M_{R} \begin{pmatrix} x \\ y \end{pmatrix}$. ...
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0answers
26 views

Easy question, hard solution: find the area about a domain in the plane?

We want to find the area of a domain with piecewisely smooth boundary by using the coordinates $(p,\theta)$ of the random line: It has been known that every straight line $\ell$ on $R^2$ can be ...
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2answers
41 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), ...
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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 ...
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2answers
20 views

Analytic geometry and definite integrals problem…

So, here's the problem: We have a parabola $y^2=2px$ and a line which is perpendicular to parabola and forms the angle $\frac{3\pi}{4}$ with x axis. I have to find the area between the parabola and ...
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1answer
156 views

A line and a plane in 3 dimensions

Line L is defined as point P(3, 2, -2) with a directional vector v <1, -1, 2>. Plane S: A $(1, 2, 1)$, B $(2, -1, 2)$, C $(0, -2, 1)$ a) When t = 2, how far away is the particle's position ...
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1answer
147 views

Intersection Point of a Line and four Planes

Let's assume a helicopter crashes into a wall after flying in a straight line: $$g : \overrightarrow {OX} = \begin{pmatrix}2\\5\\28 \end{pmatrix}+ \lambda*\begin{pmatrix}1\\\frac{1}{3}\\\frac{-1}{11} ...
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3answers
5k views

Find the equation of the plane that passes through the line of intersection of the planes…

Find the equation of the plane that passes through the line of intersection of the planes $4x - 2y + z - 3 = 0$ and $2x - y + 3z + 1 = 0$, and that is perpendicular to the plane $3x + y - z + 7 = 0$. ...
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3answers
138 views

Determining vector line equation from two planes.

Determine a vector equation of the line of intersection of the planes p1: $3x - y + 4z - 2 = 0$ and p2: $x + 6y + 10z + 8 = 0$. I know that I can find the cross product of the normals of these ...
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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) = ...
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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 ${ ...
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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 ...
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1answer
12 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 ...
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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 ...
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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. ...
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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 ...
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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 $\left(a-\frac{1}{r^2}\right)\left(b-\frac{1}{r^2}\right)=h^2$ ...
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26 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 ...
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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 ...
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2answers
238 views

Calculating spherical distance between two geo-locations

I wanted to show my nephew(16) a simple approach to calculate the distance between two geo-locations. The mathematical knowledge of a 16-year old boy is limited to simple geometrical shapes like ...
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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 ...
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1answer
27 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 ...
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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 $0.25$ ...
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1answer
524 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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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 ...
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3answers
93 views

The sum of squares of two line segments formed by a circle and coordinate axes

We have an angle of 90° so that there are 2 points A, B on each side of the angle, O is the vertex and |OA| = |OB|. On the arc AB with it's center being in O, we pick an arbitrary point P and draw ...
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1answer
85 views

Getting coordinate between two coordinates knowing the distance and latitude

That is my wall: I know the coordinates of the lower points (left and right). (X1,Y1,Z) and (X2,Y2,Z) where X is the latitude, Y longitude and Z the altitude. I want to know the another point of ...
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1answer
26 views

Distance of the Focus of an Hyperbola to the X-Axis

For a National Board Exam Review: How far from the $x$-axis is the focus of the hyperbola $x^2 -2y^2 + 4x + 4y + 4$? Answer is $2.73$ Simplify into Standard Form: $$ \frac{ (y-1)^2 }{} - ...
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1answer
199 views

Equation of a parabola-shaped toroidal tube with circular cross-sections

I need an implicit function that plots the surface that I am showing you in the picture. Everything you need is shown there. The surface is a tube in the shape of a parabola. The radius of its ...
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1answer
43 views

Prove that the line $PQ$ passes through a fixed point

A right isosceles triangle $AOB$ ($O$ being the origin), is such that when $AO$ and $BO$ are extended to points $P$ and $Q$ the relation $2AP.BQ=AB^2$ holds. Prove that the line $PQ$ passes through a ...
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384 views

Solving a system of quadratic equations

I'm facing a rather trivial problem which I seem unable to solve... Not being a mathematician (but an engineer with a bit of knack for math), I managed to formulate it in a way that seemed solvable to ...
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1answer
366 views

$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point ...
2
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1answer
460 views

Trilateration with bounds?

The problem came up doing one of my small robot experiments, the basic idea, is that each little robot has the ability to approximate the distance, from themselves to an object, however the ...