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

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1answer
78 views

Finding the points of intersection of a circle and a line

In a test (of math in arabic language) we we're asked to find the points of intersection of a circle and a line. Their equation is given. In the test I solved system of equations made of their ...
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2answers
405 views

Check my answers to the problems related to analytic-geometry

1) Find the equation of the circle of radius $2$ with center at $(3, 0)$. My answer: $\sqrt{(x-3)^2 + y^2} = 2$ 2) Find the equation of the circle of radius $\sqrt3$ with center at (-1, -2). ...
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6answers
6k views

Horizontal tank with hemispherical ends depth to capacity calculation

I am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capacity ...
2
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1answer
78 views

Compute the area of a parallelogram defined by a particular construction

I got stuck with this mathematical task. Can someone help me how to solve this problem? I need to find the F(area) value. It is kind of a thinking task Context The problem is extracted from a ...
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1answer
44 views

Standing at the center of a cube and walking halfway to a wall - field of vision

In my python programming class one of the bonus problems is this: Suppose you are located at the exact center of a cube. If you could look all around you in every direction, each wall of the cube ...
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1answer
24 views

Find the equation of parabola tangent to a line

I know how to find the equation of the line tangent to a parabola through a certain point. But how do I find the equation of the parabola from the point and the tangent line? For example, how do I ...
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1answer
376 views

closest point on a plane to another point in $\mathbb{R}^3$

Given $4$ points in $\mathbb{R}^3$: $A(0,2,4);B(-2,6,-2);C(2,-4,8);D(10,2,0)$, find the line equation $AK$ when $K$ is the projection of $D$ on the plane $ABC$. The first thing I did was find the ...
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0answers
31 views

intersecting point of two lines

The circle has R radius and and ellipse is intersecting the circle. I need to findout $x_c$ and $y_c$, which is the midpoint of the 2 intersected point of ellipse.Line 3 is the tangent of the ...
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1answer
22 views

Recurrence relation of distances between $n$-dimensional curves

I have a problem involving recurrence and euclidean distances in $n$-dimensional curves. Given the sequence of curves in $\mathbb{R}^n:$ $\{ x_{1}^2+x_{2}^2+\cdots + x_{n}^2 = 1, ...
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1answer
34 views

Calculating XY coordinates on line

I have been working on this problem for a while now and can’t figure out the solution. Hence my post on this forum. I’m trying to figure out the position of a symbol on a line. These lines are located ...
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2answers
32 views

Line passing through given points [closed]

Calculate the equation of the line through the points $(-1, 1)$, $(4, 0)$, and $(24, -4)$. the choices of answers are $y=-\dfrac15x+4/5$ $y=\dfrac15x+4$ $y=-5x+4$ $y=5x+6$
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1answer
22 views

Locus of intersection between $y= 8\lambda/(\lambda ^2 + 4)$ and $y =2 \lambda x/(4-\lambda^2)$

I have the equations $$y=\frac{4\lambda}{\frac{1}{2}\lambda^2+2}\quad \text{and}\quad y=\frac{\lambda x}{-\frac{1}{2}\lambda ^2 + 2}$$ each representing a line. I'm asked to find the locus of the ...
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1answer
50 views

What is the value of $k$? [closed]

A line has an equation of $x+5y-4=0$. If the line $x-ky-11=0$ makes an angle of $45^\circ$ counterclockwise from $x+5y-4=0$, find $k$. (If someone knows the answer, please tell me what it is and how ...
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1answer
25 views

How do I approach this geometrical problem?

For a point $P=(x,y)$ write $f(P)=ax+by$. Let $f(A)=f(B)=10$. $C$ be a point not lying on the line joining $A$ and $B$. $C^{'}$ be the reflection of $C$ w.r.t. this line. If $f(C)=15$, find ...
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0answers
42 views

Find the slope given the angle [closed]

This is the problem: find the slope of the Line 2 such that the tangent of the angle from line 1 to line 2 is -1/2? I already use the formula of $$\tan= ...
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1answer
16 views

find x in coordinates given the angle

This is the problem: if the angle from the line through $(-4,2)$and $(3,-4)$ to the line through $(-4,2) (x,3)$ is arctan 37/29 find the value of $x$? Should i use this formula: $$\tan \theta= ...
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1answer
27 views

defining a closed curve in cartesian coordinates

I am trying to implement a track in cartesian coordinates, such that X and Y coordinates are accepted and those are linearly interpolated. The problem is, I want to include circular shapes on ...
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1answer
26 views

Coordinates of a vertex of a triangle?

Here is the problem: There is a triangle with vertices $A,B,C$ in a cartesian coordinate system, where coordinates of points $A$ and $B$ and the angle $\alpha=\measuredangle ABC$ are given. The ratio ...
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0answers
18 views

Simplest way to calculate the width of a segment of a convex shape

A convex shape $C$ is cut using a a chord. What is the width of the resulting segment? This is the length of the green thick short line in the figure below: Here is my current solution: Mark the ...
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1answer
351 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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2answers
53 views

Intersection of a cone and a plane.

I need a proof that the intersection of a cone with a plane parallel to the cone's axis is a hyperbola.
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0answers
29 views

Finding the point on an ellipse most distant from a given line

$\mathrm C$onsider an ellipse with the origin as its centre, i.e., of the type $$\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$$ and a line joining two points on the ellipse. $\mathrm T$he problem is to ...
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4answers
7k views

Find intersection of two 3D lines

I have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. Please some body tell me how can I find the intersection of these lines. EDIT: By using ...
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2answers
53 views

Indefinite integral with sector of ellipse

An ellipse is given by the following equation: $$ 152 x^2 - 300 x y + 150 y^2 - 42 x + 40 y + 3 = 0 $$ After solving for the midpoint we have: $$ 152 (x-1/2)^2 - 300 (x-1/2) (y-11/30) + 150 ...
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1answer
46 views

How do you 'rotate' a polynomial?

I have a polynomial equation: $$y=(-5 \times 10^{-6} \times x^3)+(0.0004 \times x^2)+(0.0582 \times x)-0.4397$$ Is it possible to "rotate" this polynomial curve (maintaining the shape) around the ...
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1answer
415 views

Proper mapping theorem

My professor mentioned a proper mapping theorem after the name of Remmert which says: Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex ...
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1answer
21 views

Find the images of (1,0) under reflection in L?

Consider the line $$L = \{(x,y): x - 2y = 2\}$$ Find the images of $(1,0)$ under reflection in $L$? Thanks in advance.
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2answers
27 views

How to interpret the equation of a line in 3D through two points, when there are $0$s in the denominator? [closed]

If $A=(0,0,0)$ and $B=(1,0,0)$ are two points of a line in three dimensions, I think its equation should be $$\frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0}\tag1$$ according to the formula ...
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1answer
25 views

doubt with direction angles

Is it possible for a 3D vector to be drawn with the direction angles of $\alpha=45^\circ$ and $\beta=45^\circ$ ? if yes what is measure of $\gamma^\circ$? I calculated $\cos^2(45^\circ ...
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0answers
18 views

Proof in analytic geometry using vector multiplication

Let us have a triangle $\Delta ABC$. $H$ is the intersection of heights if and only if $$ \overrightarrow{HA}\cdot \overrightarrow{HB} = \overrightarrow{HB}\cdot \overrightarrow{HC} = ...
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1answer
26 views

Algebraic step on a trig expressiom in linear algebra

$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$ $$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 ...
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0answers
32 views

Proof in analytic geometry

Let us have a triangle $\overset{\Delta}{ABC}$. $H$ is the intersection of heights if and only if $\vec{HA}.\vec{HB}=\vec{HB}.\vec{HC}=\vec{HC}.\vec{HA}$ I think that it has to deal with vector ...
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0answers
58 views

Find max distance from $(0,0)$ to line defined on ellipse.

I have got a following problem : $E = \{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 \}$ $N$ - line (normal) perpendicular to E at $(x_0,y_0)$ Find max $dist(N,(0,0))$ So I am starting with attempt to ...
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2answers
37 views

Questions about elipse

Given the center of an elipse and three of its points, is this elipse completely determined? What is the easiest way to show that five points of an elipse are enough to determine the elipse?
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1answer
750 views

Find equations of the ellipses given conditions on the directrices, foci, and vertices

The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation: with distance between directrices $27$, and between foci $3$; with a focus at $(-\sqrt{13},0)$ ...
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2answers
27 views

Find intersection points of a line with a circle, and the equation of another circle passing through those points [closed]

If the line $x=2y$ meets the circle $x^2+y^2-8x+6y-15=0$ at points $P,Q$, find the co-ordinates of $P$ and $Q$ and the equation of the circle passing through $P,Q$ and at the point $(1,1)$. Could I ...
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1answer
21 views

How to find a point at a certain distance to other points on the same line

Assuming the points A(x1,y1) and B(x2,y2) and distances between AB (d1) and AC (d2) are known. How can I find the point C(xp,yp)? Actually it has a trivial solution, writing the distance equation 2 ...
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2answers
21 views

Find a Cartesian Equation for the Plane Satisfying Those Properties

Find the Cartesian equation of the plan parallel to j and passes through the intersection of the planes described by the equations x + 2y + 3z = 4, and 2x + y + z = 2. I was able to get the ...
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0answers
16 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
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1answer
67 views

Triangle, Circle Problem

What is the area $\triangle DEF$ ? I solved it using analityc geometry. I want to see if there is way to solve it using plane geometry. I did it: $x^2+y^2=400$ $(x+10)^2+y^2=100$ I found the ...
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0answers
15 views

Plane and symmetrical lines

I need to solve this problem (sorry for bad english) I have plane $\pi$ and line $p_1$ intersecting $\pi$ in point $P$. Then I find line $p_2$ symmetrical to line $p_1$ where $\pi$ is plane of ...
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1answer
21 views

How to show existence of an orthogonal map?

I want to show that the following holds: Let $x,y\in \mathbb{R}^n\setminus\{0\}$ be given and such that $\|x\|=\|y\|$. There is an orthogonal map $T$ such that $Ty=x$ (a rotation). How could one ...
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5answers
355 views

Circle radius as variable

I am confused. How is $y^2 + x^2 =3x$ a circle? Can someone please help me try to understand why the above a circle, or is it just a typo?
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2answers
23 views

Find the equation of the plane that contains:

Find an equation for the plane containing the lines $$x = 5y = \frac{z + 1}{4}$$ and $$\begin{cases} x = t \\ y = 2t\\ z = 6t − 1 \end{cases}.$$ I know that finding two points will allow me to find ...
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1answer
44 views

Finding a line through 4 other lines!

This one's probably easy, but I'm dreadfully stuck and can't seem to figure out a decent method. I have the following lines: $$a: \vec{x}(\lambda)= \left( \begin{array}{ccc} 4 \\ -2 \\ -2 ...
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1answer
338 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
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4answers
54 views

Show that if an ellipse and a hyperbola have the same foci, then at each point of intersection their tangent lines are perpendicular.

I have to show that: If an ellipse and a hyperbola have the same foci, then at each point of intersection, their tangent lines are perpendicular. So I know that if I prove it for one of the ...
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0answers
11 views

Equivalence of definitions for a conic

I have to prove that these two definitions for the eccentricity of a conic $C$ are equivalent: Ratio between the distance of the points $x$ in $C$ to $f$ its foci and $l$ its directrix. Ratio ...
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2answers
23 views

Showing that normal line passes through a point.

I need to show that a line passes through a point. How should I go about doing this? The question is: Let $L$ be the normal line at $(1,1,1)$ to the level surface of $f(x,y,z) = x^2 - z$ that ...
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2answers
355 views

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?