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

learn more… | top users | synonyms (1)

0
votes
2answers
31 views

How to derive formula for focus of a parabola?

I understand how to obtain the formula for the vertex of a formula, $ y= a(x-h) + k $ where $ h=-b/2a$ and the vertex is $(h,k)$. However I don't know how to get to $(h,k+1/4a)$. Could someone please ...
0
votes
1answer
21 views

Ellipse - relation between b such that $F_1P \perp F_2P$

Consider the ellipse $\displaystyle \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$ with foci $F_1 (-e, 0)$ and $F_2 (0, e)$ (where $e$ is the linear eccentricity). What is the relation between $a$ and $b$ so ...
0
votes
0answers
15 views

Homogenization of Equations

Say there are two equations: $3x^2+mxy-4x+1=0$ and $2x+y-1=0$. I have to find possible values of $m$ for which lines joining the points of intersection of above two equations are at right angles. I ...
0
votes
0answers
10 views

Spherical coordinates after rotations in 3D

It's straightforward to derive rotation matrices in 3D space around the x, y and z axes. Those matrices give the new coordinates x', y' and z' in terms of the old components x, y and z and the angle ...
1
vote
2answers
50 views

Can $n$ circles be drawn such that all have a common intersection but no two intersect individually

I was fiddling with plane geometry when a question came into my mind: Can $n$ circles ($n \ge 3$, $n \in \mathbb{N}$) be drawn such that: $C_1 \cap C_2 \cap C_3 \cap \ldots \cap C_n \not = ...
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 ...
0
votes
0answers
17 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
23 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 ...
3
votes
0answers
28 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 ...
1
vote
1answer
33 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 ...
5
votes
0answers
62 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 ...
0
votes
1answer
15 views

Normal vector between two parallel lines [closed]

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?
6
votes
2answers
100 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| = ...
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
0answers
38 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} ...
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
0answers
9 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
27 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 ...
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?
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
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
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 ...
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
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? ...
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
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 ...
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 ...
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 ...
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 ...
0
votes
1answer
18 views

Find the coordinates of E, G and H, and calculate the area of shape OFEH

Currently I am looking at a graph of a circle. The diameter is y=2x+3 Tangent at point E cuts the x-axis at F (12;0) 1. find the coordinates of E 2. find the coordinates of G and H (H being the ...
0
votes
1answer
11 views

calculate the value of P is the points A(6;5), B(3;2) and C (2p;p+4) are co-linnear

Also honestly have no clue whatsoever. I have tried jotting down a graph and just finding the differences between A and B and minusing them from B to create C. I know this is completely wrong! Please ...
-2
votes
1answer
26 views

In a triangle $ABC$ with $A=(1,3) ,B =(q,0), C =(p,-4)$ [closed]

Let $A=(1,3),B =(q,0), C =(p,-4)$, with $p>0$, the slope of $AB$ is $+45^\circ$ and $AC= \sqrt{50}$. Determine the gradient of $AB$ Calculate the equation of the line $AB$ Calculate the value of ...
-1
votes
0answers
11 views

Analytical geometry graph P (-3;0) s (0;2) R (2;-1) Q (-1;-3)

show that PR and QS bisect each other. Honestly have no idea how to do this. Basically the shape looks like a square and the midpoint coordinates of PR are (-1/2; -1/2)
1
vote
0answers
41 views

Find the equations of the lines of greatest slope and least slope

Find the equations of the lines of greatest slope and least slope on the plane $3x-4y+5z-5=0$ drawn through the point $(1,2,2)$ given that the plane $4x-5y+6z-6=0$ is horizontal. I do not need ...
-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 $\min\{a+b\}$
1
vote
3answers
35 views

Given three coordinates (a,b,c), (d,e,f), and (l,m,n), what is the center of the circle in the 3D plane (h,k,i) that contains these three points.

I have tried the following: $$(a-h)^2+(b-k)^2+(c-i)^2=r^2$$ $$(d-h)^2+(e-k)^2+(f-i)^2=r^2$$ $$(l-h)^2+(m-k)^2+(n-i)^2=r^2$$ Subtracted equation 2 from 1, equation 3 from equation 2, and equation 3 ...
0
votes
1answer
28 views

Proving a vector bisect two other vectors

How can I prove the vector: $$ \vec{w}=|\vec{u}|\vec{v} + |\vec{v}| \vec{u} $$ bisects the angle between the vectors $\vec{u}$ and $\vec{v}$ ? I have trying using the scalar product, but it does not ...
0
votes
0answers
15 views

Understanding distance between point and line via infimum

The distance between point and line independently on metric is defined by $$d(X, l) = \inf\{d(X, Y)|Y\in l\}.$$ I have troubles understandning how this infimum works. Can someone please give me an ...
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: ...