# Tagged Questions

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

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### Analytic Geometry: One sheeted hyperboloid

Good afternoon! I have a question about analytic geometry. I don't actually know if the answer is quite simple, and I missed something while revising, or if it is actually more complicated than I ...
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### Determination of a volume.

Consider, in the Cartesian plane, the square Q having vertices in the points $(-1, 0), (1, 0), (0, -1)$, and $(0, 1)$. The sections of a solid with planes orthogonal to $y=0$ are squares having two ...
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### Rotation around a line which is determined by two points in 3D space

If we have three points like $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$ and $C(a,b,c)$. Then, $A$ and $B$ determines a line like $l$. After that, we rotate $C$ around $l$ by $\omega$ degree (anti-clockwise). ...
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### A book on analytic geometry

It's easy to find good recommendation for books here for any subject other than analytic geometry ,therefore I'd like to ask for any suggestion of analytic geometry books ,the only charactrestic I'm ...
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### A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
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### Equation of rectangle

I need equation of a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? for example equation of ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
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### Proof of Descartes' theorem

I came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any ...
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### Given that the graph of $f$ passes through the point $(1, 6)$ and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$, find $f(2)$.

As in the title - we assume that the graph of $f$ passes through $(1,6)$ (i.e. $f(1) = 6$) and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$ and we are asked to find $f(2)$. How does ...
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### What is the basic idea of homogenisation of an equation?

I do get that when you are homogenising it makes it in an equation of pair of straight lines passing through origin but what is its actual point and its applications?
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### Lines joining origin to points of intersection of two conics

If the lines joining origin and point of intersection of curves $$ax^2+2hxy+by^2+2gx=0$$ and $$a_1x^2+2h_1xy+b_1y^2+2g_1x=0$$ are mutually perpendicular, then prove that $$g(a_1+b_1)=g_1(a+b)$$ How ...
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### Coordinate Geometry of circles; Radical Axis question

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to the circle with center at $(2, 1)$, then the radius of the second circle is? Apparently the solution is $3$, with the cryptic ...
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### Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
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### Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right)$$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
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### Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
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### Three planar vectors $x,y,z$ such that $x$ is orthogonal to $y + z$ and $z$

Let $x$ be a non-zero vector, orthogonal to vectors $y + z$ and $z$, with $x, y, z \in \mathbb R^2$. Prove that $y$, $y - z$ and $z - y$ are orthogonal to $x$ and parallel to $z$. To prove they are ...
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### intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
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### Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
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### Doubts on locus and its equation

"Find the equation to the locus of a point which is collinear with points $M(a,0)$ and $N(0,b)$." The answer is $- x/a + y/b$. How I tried to find the solution: $P$ is a point whose assigned ...
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### Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
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### Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
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### Number of non zero integer values of $k$ for which the points ($k,k^2)$ lies inside the triangle formed by the given three lines

Problem : Number of non zero integer values of $k$ for which the points ($k,k^2)$ lies inside the triangle formed by the lines $11x+6y+14=0$, $9x+y-12=0$, $2x+5y-17=0$ (a) $0$ (b) $2$ (c) $3$ ...
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### Line not intersecting circle, maximum value of expression involving radius

If line $y+x=2$ do not intersect any member of circles $x^2 + y^2 -ax = 0$ at two distinct points where a is parameter, then maximum value of $|a + 4|$. My try: Since the line does not intersect ...
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### Line parallel to plane

See if the line e is parallel with to the plane $α$. If not, find the intersecting point. \begin{align} α: & \quad \quad x-3y+z+1=0 \\ e: & \quad \{x+y-z=3, 2x-y-4z=3\} \end{align}
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### Equation of parabola, tangent at vertex [closed]

Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $(2,3)$ is the focus of the parabola, then find the equation of tangent at the vertex. Thanks. My thoughts: Can't figure out anything :(
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### Determine Center Point based on 2 separate elipses

First timer here. I've been digging back into my good old maths days but am extremely rusty (beyond belief). I got a really tricky question that i want to determine formula for so that my mate can ...
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### Find the line through $(-1,4)$ for which the distance to $(6,3)$ is 5

This is the question: Find the line through $(-1,4)$ for which the distance to $(6,3)$ is $5$ The answer is: $y-4=-4/3(x+1)$ and $y-4=3/4(x+1)$ I do not know how to get this answer. ...
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### Definition of (hyper)planes

I know the definition of a plane to be: $(r-r_0)\cdot n = 0$ where $n$ is the vector perpendicular to the plane, $r$ the vector to a given point and $r_0$ the vectors to the points which constitute ...
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### Showing that a circle is “tangent” to the $x$-axis if and only if $\left|k\right| = r$.

The problem is this: to show that a circle of radius $r$ and center $(h, k)$ intersects the $x$-axis at exactly one point if and only if $\left|k\right| = r$. Using geometrical intuition, this ...
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### Intuition for a certain tensor product.

Tensor products occur in lots of places and until recently I thought I understood them at least reasonably well. During the past few weeks, however, I've attended several talks where the tensor ...
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### Find the equation of a circle which is tangent to $y$-axis at a given point and cuts a chord of given length on the $x$-axis

How to find the equation of the circle which touches $y$ axis at $(0,3)$ and cuts a chord of length $8$ on the $x$ axis? It should look like this: My approach: Since the circle touches $y$ ...
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### Can asymptotes be curved?

When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form $y=a$) and vertical ones ( of form $x=b$). I was then shown oblique asymptotes-- slanted ...
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### How can I convert the following parametric equation to cartesian equation?

\begin{align} x&=\left(1 + \frac{1}{\,\sqrt{\,2t^{2} - 4t + 4\,}\,}\right)t\ -\ 2 \\[3mm] y&=\left(1 - \frac{1}{\,\sqrt{\,2t^{2} - 4t + 4\,}\,}\right)t\ +\ \frac{2}{\,\sqrt{\,2t^{2} - 4t + 4\,}...
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### How can I find the volume of this prism and points B, C, D and F?

In the triangular prism, A = (0, -1, 1), E = (0, -3, 0). C and D belong to line s: x - 1 = y = y - z. How can I find the prism's volume and the coordinates of B, C, D and F points?
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### Find a specific vector equation of a line that divides a angle in half.

I've been studying a little geometry on my own, and I just recently stumbled on this problem, that I'm unable to answer: Given the points A=(2,-1), B=(5,4) and C=(-7,8), find a vector equation of a ...