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

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6
votes
2answers
91 views

Locus of a point on a fixed-length segment whose endpoints slide along orthogonal lines

Suppose we have some segment $AB$ of constant length that slides in such a way that its endpoints are moving along orthogonal lines. Let $P$ be a point in the segment so that $|AP| = a$ and $|PB| = ...
2
votes
2answers
84 views

Find the equation of the circle which cuts the circle $x^2+y^2+2x+4y-4=0\;$ and the lines $xy-2x-y+2=0\;$ orthogonally

The equation of the circle which cuts the circle $x^2+y^2+2x+4y-4=0\quad$ and the lines $xy-2x-y+2=0\quad$ orthogonally, is $a.\quad x^2+y^2-2x-4y-6=0\;$ $b.\quad x^2+y^2-2x-4y+6=0\;$ $c.\quad ...
0
votes
1answer
35 views

Vector Distance

let there be a line L: $\frac{x-1}{2}= \frac{y+1}{3}= \frac{z}{1}$ and a plane: $2x-y-z=5$. With this given data find: a line L1, such that L1 is parallel to L, is in P, and the distance between L and ...
3
votes
0answers
27 views

Cosine Inequality, Geometric interpretation in the complex plane

The following identity was given as an exercise in the course notes for a complex analysis course. I am able to solve it (the proof is given below), but am unsure of the geometric interpretation of ...
0
votes
0answers
16 views

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. Centroid of $\Delta ABC$ lies on $y=3x-4$, then the locus of $D$

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. If centroid of $\Delta ABC$ lies on $y=3x-4$, then what is the locus of $D$? I did try a couple of things, but I honestly ...
0
votes
1answer
14 views

How to determine if a point lies in this particular convex region?

I have a family of hyperplanes which do not contain the origin: \begin{eqnarray} a_{11}x_1+a_{12}x_2+\dots+a_{1n}x_n &=& k_1\\ a_{21}x_1+a_{22}x_2+\dots+a_{2n}x_n &=& k_2\\ ...
0
votes
4answers
22 views

Check if a given coordinate lies in path of a ray (coordinate geometry)

As shown in the image I have two known coordinate pair A and B and few other known coordinate pairs (RED blob) on the graph. I need to know if any of the other given coordinates fall in line of the ...
1
vote
1answer
31 views

Where i am going wrong in finding normal to curve?

The question is Find the perpendicular distance between the normal to the curve $$x=a\cos t+at\sin t, y=a\sin t-at\cos t$$ and the origin. Equation is given in parameterized form. My attempt ...
-1
votes
1answer
24 views

Question on circles…

If three circles with radii ${3}$,${4}$,${5}$ touch each other externally at points P,Q and R,then the CIRCUMRADIUS of ∆PQR is...?? My attempt i think that the let the point of the common ...
2
votes
2answers
687 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
5
votes
0answers
57 views

Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
6
votes
1answer
831 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 ...
0
votes
1answer
15 views

Normal vector between two parallel lines [on hold]

Is there a way to calculate the normal vector of two parallel lines, without calculating the length or the points?
1
vote
0answers
64 views

Will the boy outwit the teacher in this way? [duplicate]

In the book, Solving Mathematical Problems: A personal perspective (written by Terry Tao), he discusses a problem named (on Analytic Geometry Chapter, page 79): Problem 5.4 (Taylor 1989, p. 34, ...
0
votes
1answer
17 views

How to proove that foot of perpendicular drawn from focus to any tangent of an ellipse lie on auxillary circle?

One way is to find the foot of perpendicular and directly putting it into the equation of auxiliary circle. But that is quite a lengthy proof, is there any other short method to prove this property?
0
votes
0answers
17 views

Finding the equation of a locus [closed]

Find the equation of the locus of a point which moves so that its distance from (4,-3) is always one-half its distance from (-1,-1) . How to solve it? XD
0
votes
1answer
18 views

Can a line parallel to axis of parabola also represent tangent at a point along with the one whose slope is found using calculus?

Consider a parabola with the equation $y^2=4x$ its axis is the x-axis and vertex is (0,0) and focus at (1,0). Consider any point on the parabola say (4,4). Now we define tangent at this point as a ...
2
votes
1answer
30 views

Partition a triangle into equal areas

A piece of wooden board in the shape of an isosceles right triangle, with sides $1$,$1$, $\sqrt{2}$ is to be sawn into two pieces. Find the length and location of the shortest straight cut which ...
0
votes
1answer
43 views

$ax^2+by^2+2gx+2fy+2hxy+c=0$ : Understanding the equation

Given any second degree equation in $x$ and $y$, $ax^2+by^2+2gx+2fy+2hxy+c=0$ is it possible to find out the centre and/or the axis of the conic section it represents? What information can I ...
0
votes
3answers
30 views

PARABOLA : Problem

Find the equation of line touching both the parabolas $$ x^2=-32y.......(1)$$ $$ y^2=4x.........(2) $$ i have equated slopes of both the parabolas and applied the condition that all the points on ...
0
votes
0answers
37 views

HYPERBOLA : Problem [duplicate]

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} -\frac{y^2}{b^2} = 1$ whose centre is $C(0,0)$ are such that $CP$ is perpendicular to $CQ$ , $a<b$ , then prove that $$\frac{1}{(CP)^2} ...
-1
votes
2answers
23 views

Finding radius of a circle given $3$ points and center. [closed]

Show that the points $(-2,5), (2,-1)$ and $(4,-1)$ all lie on a circle whose center is at $(1,2)$. Find the length of the radius.
0
votes
1answer
10 views

Eccentricity of a hyperbola given the angle between the x-axis and its asymptote

I need to find the eccentricity of a hyperbola whose asymptote makes an angle $\alpha$ with the $x$-axis. So, I take the case where the transverse axis will be horizontal, $i.e.$ ...
0
votes
1answer
26 views

If center of rhombus is $(\pi, e)$. FInd the equation of diagonal

Two sides of rhombus are parallel to $3x+4y+17=0$ and $4x+3y+16=0$. Center of rhombus is $(\pi, e)$, find the equation of its diagonal. Is data in this question sufficient to find required diagonal?
3
votes
1answer
391 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 ...
0
votes
0answers
8 views

A question about the normal form of a hipercuadric

What would be the analogue of the notion of normal form of a hipercuadric if we work on Q? Please,could you help me?
0
votes
1answer
37 views

Ellipse and chord length

There is a analytic geometry problem: In the ellipse $\frac{x^2}{4}+y^2=1$, segment $AB$ is a chord and $AB=\sqrt{3}$, find the maximum and minimum area of $\triangle AOB$. My progress Assume ...
1
vote
1answer
26 views

Finding parametric equations of rectangular equation

Is there a general process to follow when finding the parametric equations of a normal rectangular equation ? I know that one rectangular equation might have many parametric equations, but are there ...
3
votes
1answer
33 views

Let $f(x)=x^5$. For $x_1>0$, let $p_1=(x_1,f(x_1))$.Draw a tangent at the point $p_1$

Let $f(x)=x^5$. For $x_1>0$, let $p_1=(x_1,f(x_1))$. Draw a tangent at the point $p_1$ and let it meet the graph again at point $p_2$. Then draw a tangent at $p_2$ and so on . Show that , the ratio ...
3
votes
2answers
436 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, ...
0
votes
1answer
489 views

deriving formula for reflection over y=mx+b using dot product

So, I know that the formula for a generic point is $$\left(\frac{1-m^2}{1+m^2}x + \frac{2m}{1+m^2}(y-b), \left(\frac{2m}{1+m^2}\right)x - \left(\frac{1-m^2}{1+m^2}\right)(y-b)+b\right)$$ when you ...
0
votes
1answer
655 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 ...
0
votes
1answer
27 views

Two coordinates, two angles, and the third coordinate

Let $A$, $B$ and $C$ be points on a two-dimensional coordinate system. Assume $A=(0,1), B=(0,5)$, angle $\alpha$ of $A$ is 47 degrees, and angle $\beta$ of $B$ is 80 degrees. Calculate the ...
1
vote
0answers
21 views

Equation of a Plane

I realize this may be VERY low level for this forum. I'm practicing for an exam and I just want to verify an answer because I do not have the solutions for this practice test. The question is: Find ...
1
vote
1answer
28 views

$A(1,1,1)$, $B(2,1,2)$, $C(3,2,1)$ and $D(2,3,2)$ form a tetrahedron. If $ABC$ is the base, then what is the height?

$A(1,1,1)$, $B(2,1,2)$, $C(3,2,1)$ and $D(2,3,2)$ form a tetrahedron. If $ABC$ is the base, then what is the height? I found out of the equation of the plane containing A, B and C. It is $$-x + 2y +z ...
1
vote
2answers
34 views

Distance between incenters and excenters

In a triangle ABC,if $(II_1)^2+(I_2I_3)^2=\lambda R^2$,where I denotes incenter,$I_1,I_2,I_3$denotes centers of the circles escribed to the sides BC,CA and AB respectively and R be the radius of the ...
1
vote
1answer
19 views

The expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$.

Using vectors I tried obtain the expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$. The point of intersection is ...
1
vote
0answers
22 views

Getting topological objects from the “cube” of $T^3$

One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$. To get ...
0
votes
1answer
29 views

How to calculate a point between two angled lines based on distance from the lines?

Please take a look at the picture below for the diagram reference: I am trying to calculate the point where it is perfectly 3.3 cm vertically from the 44.52 cm line AND 5.5 cm horizontally from the ...
1
vote
1answer
23 views

Changing the side of a triangle without changing area?

$\triangle ABC$ has vertices $A=(8,2)$, $B=(0,6)$ and $C=(-3,2)$. Point $C$ can be moved along a certain line with points $A$ and $B$ remaining stationary so that the area of $ABC$ will not change? ...
1
vote
0answers
60 views

Show that the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes

Show that the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$ is $(-7,-1,-9)$. ...
0
votes
1answer
19 views

Verify that $R_{(a,b)}\subset D$.

Let $(a,b)$ be any point in the disk $D = \{(x,y): x^2 + y^2 < 1\}$. Put $r=\sqrt{a^2 + b^2}$. Let $R_{(a,b)}$ be the open rectangle with vertices at the points $\left(a\pm\frac{1-r}{8}, b ...
-2
votes
0answers
23 views

Suppose f(z)= u(x,y) + iv(x,y) is a nonconstant analytic function

please I need someone to help me to solve this problem. Indeed I could not understand what the question wants? I am confused thanks. Suppose f(z)= u(x,y) + iv(x,y) is a nonconstant analytic function ...
1
vote
2answers
51 views

What is condition for second degree equation to represent a pair of straight lines?

According to my text the necessary and sufficient condition for a general equation of second degree i.e. $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a pair of straight lines is that 1) the ...
1
vote
2answers
47 views

How can I solve this line & plane intersect question and verify the given answer? [closed]

Find an equation for the plane that passes through the point $(3,2,1)$ and contains the line of intersection of the planes with equations $x+y+z=3$ and $x+2y+3z=6$. The given answer from the key is: ...
2
votes
5answers
2k views

How to find coordinates of reflected point?

How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? Example Question: If P is a reflection (image) of point (3, -3) in the line $2y = ...
2
votes
1answer
30 views

Find the co-ordinates of the point on the join of two points which is nearest to the intersection of two planes

Find the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$. Please give me an ...
4
votes
5answers
1k views

3D coordinates of circle center given three point on the circle.

Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?
0
votes
1answer
26 views

What are the coordinates for the center of the second circle? (Full question in body)

Full Question:A circle has its center at (6,7) and goes through the point (1,4). A second circle is tangent to the first circle at the point (1,4) and has one-fourth the area. What are the coordinates ...
-2
votes
3answers
64 views

Find minimum of $a+b$ under the condition $\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$ where $m,n$ are fixed arguments

Assume $m,n \in \mathbb{R}$ is fixed. And $a,b(a>b>0)$ satisfied the equation $$\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$$ Find $\min\{a+b\}$