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

learn more… | top users | synonyms (1)

0
votes
1answer
28 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
74 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
19 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 centre)...
0
votes
1answer
13 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
27 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
vote
0answers
65 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 the ...
-2
votes
3answers
83 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
40 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
34 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
17 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
49 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: $...
0
votes
1answer
56 views

How to find whether equation of angle bisector represents the obtuse or acute angle bisector of two given straight lines?

Two lines: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are given. I know that the equation of its bisectors is ${a_1x + b_1y + c_1 \over \sqrt{(a_1^2 + b_1^2)}} = \pm {a_2x + b_2y + c_2 \over\...
1
vote
1answer
49 views

When does $ax+by+c=0$ represents a family of straight lines passing through a fixed point?

a first degree linear equation $ax+by+c=0$ represents a family of straight lines passing through a fixed point if and only if there is linear relationship between a,b and c? How can we prove this? ...
1
vote
0answers
27 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
3answers
32 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 ...
-1
votes
1answer
19 views

How are trigonometric ratios function of interior angles in a right angled triangle?

How can one assume that the ratio altitude/hypotenuse is a function of angle. For a general right-angled triangle--->Let: Hypotenuse$=c$ Altitude$=a$ Base$=b$ and angle opposite ...
3
votes
0answers
31 views

Find the radius of the circle for given conditions

A circle with center at origin passes through three points $P$, $Q$ and $R$ with the line segment $PQ$ as its diameter along $x$-axis. A line passes through $P$ intersects the chord $QR$ at point $D$. ...
0
votes
1answer
19 views

Find the gradient of lines joining the following pair of points.

If, $Gradient = \frac{(y_2-y_1)}{(x_2-x_1)}$ And, $(x_1,y_1),(x_2,y_2) = (p+3, p-3), (3p+4, p-5)$ Then, $(y_2,y_1) = ((p-5)-(p-3))$ $=((p-5)-p+3)$ $=(p-5-p+3)$ $=(-2)$ And, $(x_2,x_1) = ((...
2
votes
0answers
49 views

Ring of germs of holomorphic functions at $0\in \mathbb{C}$

So I've been reading the book and they used a induction proof where they just state that for the base case the ring of germs of holomorphic functions on $\mathbb{C}$ is Noetherian. I looked at other ...
1
vote
1answer
18 views

Perturbation of tangent ball

As picture below, $A$ and $B$ are two balls, $\partial A\bigcap \partial B=\{k\}$, and $B$ contains $A$. How to show that $$ \forall h\in \partial B,\exists ~\varepsilon > 0 ~st~ A\subset B+\...
0
votes
1answer
27 views

if I know a point nearest the zero of a polynomial can I tell which zero it is? (finding intersect of $f(x)$ with a line)

I have a function $f(x)$ and two points $p_1$ and $p_2$. What I need to find is the point where $f(x)$ and the line defined by the two point intersect. I know what $f(x)$ is, $f(x) =\dfrac{c_0+c_2x+...
3
votes
1answer
33 views

Question about the Jacobian of a function

Let $f:U\rightarrow V$ , $U$ and $V$ open subsets of $\mathbb{R}^2$, be a smooth function. Let $Jf_p$ be the jacobian of $f$ in the point $p\in U$ and set $M_p:=\sup\{|df_pv|:\|v\|=1\}$ and $m_p:=\inf\...
0
votes
0answers
13 views

Requesting formulas about circum-circles

With the co-ordinates of the three vertices of a triangle given, we have nice looking formulas for the centroid and in-center. Do we have the same kind of formula(s) for the circum-circle or circum-...
2
votes
3answers
51 views

Why the Cartesian equations, are called “Cartesian”?

I've been studying analytic geometry and I'm wondering "Why the Cartesian equations, are called 'Cartesian,'" I know that the name is from the René Descartes philosopher. But in that one case why is ...
1
vote
1answer
18 views

Equation of a subspace given basis

Suppose we have a subspace expressed as the linear combination of two vectors (basis): $S = x * (3, -3, 1) + y * (5, 1, 3)$ How can I find the equation for the subspace (in this case, a plane ...
3
votes
1answer
81 views

A parallelogram between two points on a hexagonal lattice containing all the shortest paths

For any two points on a hexagonal grid with integer coordinates there is a unique parallelogram which contains all of the shortest paths (in terms of taxicab norm) between these points. See the ...
0
votes
1answer
26 views

Number of lattice points in triangle formed by x-axis, y-axis and given line

Given a line $ax+by=c$ where $a,b,c$ are positive integers. Is there any formula to find the number of points inside the triangle formed by this line, $x$-axis and $y$-axis? Points on the boundary ...
0
votes
2answers
35 views

Find closest Point to Another Point

How do I find the closest point to $(2,2)$ on the line CD, if C is $(3,2)$ and D is $(5, 3)$? How would I solve using linear algebra? Does it involve cross product and distance? Not sure how to solve
1
vote
0answers
18 views

Cohomology of $\mathcal{O}^*$ and projection map

Suppose $X$ is a complex manifold and $T$ a complex space (or complex manifold maybe) and let $\pi:T\times X \rightarrow T$ denote the projection. What are sufficient conditions on $X$ that make $$H^2(...
0
votes
1answer
28 views

2D coordinates of rotating a “bent line”?

I have this problem, when I am given a point A an an XY plane, and I need to find the coordinates of a point B that is of a constant distance of my point A, and my OAB angle is fixed (O being the ...
0
votes
1answer
27 views

A question about affine spaces

Are there affine spaces that contain subsets that aren't closed to affine combinations of three points? This is a surprising question. I think that exists that kind of affine spaces,but I don't know ...
2
votes
3answers
33 views

$P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii

If $P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii. $\angle PSS'=\alpha$ and $\angle PS'S=\beta$, then prove that: $$ \tan\left(\frac{\alpha}{...
3
votes
2answers
54 views

Best Fitting Pipe in parabolic trench

A work crew is digging a pipeline. The cross section of the trench is in the shape of the parabola $y = x^2$. The pipe has a circular cross section. If the pipe is too large, then the pipe will not ...
0
votes
1answer
22 views

find the coordinates of the point that divides the join of A(-1,-7) & B(1,2) internally, in 2:1.

What I wanted to ask was that after finding the coordinates of the point my answer was (1/3, -1) now since the ordinate is -ve doesn't that make this an external division? How can it divide the line ...
1
vote
4answers
56 views

Coordinate Geometry: Are there enough information to find out the coordinates?

Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and $|AB|=|AC|...
1
vote
2answers
29 views

Rotation of conics [duplicate]

How to rotate a conic by an determined angle? Could someone give me the step by step? (I know how to rotate the coordinate system by that formula \begin{align} x &= x'\cos(a) - y'\sin(a) \\ y &...
3
votes
1answer
59 views

A mirror focusing beams at one point

How can I find a shape of a mirror which focuses all parallel beams in one point? I tried to do it in this way: The mirror must be symmetric hence I assumed it has a center in the point $(0,0)$. The ...
0
votes
2answers
74 views

Get the four corners of a rectangle

I have a boundary given ($xMin$, $yMin$, $xMax$, $yMax$) and the two points of a reference line of a rectangle. The begin point is at $(x_b, y_b)$ and the end point is at $(x_e, y_e)$. This reference ...
1
vote
2answers
25 views

Polar Equation to Rectangular

$$r=\frac{9}{4 \cos θ − 3 \sin θ}$$ How do I do this? (Equation is in polar form.) I have already tried to do this, but I don't know how to finish it.
-2
votes
1answer
47 views

Give a geometrical interpretation of the intersection of the planes with equations [closed]

Give a geometrical interpretation of the intersection of the planes with equations \begin{align} &x + y − 3 = 0\\ &y + z + 5 = 0\\ &x + z + 2 = 0 \end{align} what is a geometrical ...
1
vote
3answers
28 views

Through the point $A(4,5)$ a line is drawn.

Through the point $A(4,5)$ a line is drawn inclined at $45°$ with the $+ve$ X - axis. It meets $x+y=6$ at the point $B$. Find the equation of $AB$. My solution.. Equation of $AB$ $$(y-y_1)=m(x-x_1)...
1
vote
2answers
66 views

Curve equidistant to sine and cosine.

If I have the sine and cosine curves plotted, what would be the formula of the curve that is equidistant to both curves? Here's a picture of how it looks like. The original question comes from a ...
0
votes
2answers
77 views

Geometrical interpretation of solving a $3 \times 3$ system of equations

Solve the following system of equations and give a geometrical interpretation of the result. \begin{align*} x + y + z &= 6\\ 2x + y − 3z &= -5\\ 4x − 5y + z &= −3 \end{align*} I know that ...
2
votes
1answer
38 views

Plane $3x + y - z= 4$ touches the ellipsoid $2z^2 = \sqrt7(1 - 2x^2 -y^2)$

Show that the Plane $3x + y - z= 4$ touches the ellipsoid $2z^2 = \sqrt7(1 - 2x^2 -y^2)$ My attempt: First I tried to convert the equation of ellipsoid in general form and then further applying the ...
2
votes
1answer
143 views

condition for cones to be reciprocal

Question : Show that the cone $$ax^2 + by^2 + cz^2 - cxy - ayz - bzx = 0$$ is the reciprocal of the cone $$(a^2 - bc)x^2 + (b^2 - ac)y^2 + (c^2 - ab)z^2 - 2(a^2 + bc)yz - 2(b^2 + ac)zx - 2(c^2 + ab)xy ...
0
votes
1answer
38 views

Distance between two Polar-Coordinates

I choose two Points in Berlin with the coordinates: 1: lat: 52.511206 long: 13.546486 2: lat: 52.527501 long: 13.319206 With an online tool I got the ...
0
votes
1answer
34 views

Finding the Locus of Circumcentre

Let $P$ be a point on circumcircle of $\Delta ABC$, where $A=(3,4), B=(-3,4), C=(4,3)$. Let feet of perpendicular from $P$ to $AB$ and $AC$ be $Q$ and $R$, respectively. Then locus of circumcentre of $...
1
vote
2answers
54 views

How to find whether a point lies on a line which is in parametric form?

Does the point $(1,8,3)$ line on the line with parametric equation: $$x = 5 + 2t$$ $$y = 2 + 6t$$ $$z = 1 + 3t$$ I know how to solve if they give me a equation of a plane and ask whether the ...
0
votes
1answer
13 views

A problem about affine spaces

Let A be an affine space,dim(A)=4. P,Q are planes from A. If dir(P)!=dir(Q),then P and Q are disjoint. Is this proposition true or false? I know that two planes are parallel if they are disjoint ...
1
vote
3answers
48 views

A question about an equation of a plane

Let $A=(1,3,1)$; $B=(1,1,1)$; $C=(2,0,1)$; $D=(1,-2,3)$. Determine the equation of a plane that passes through $D$ and is parallel with $(ABC)$. I know the fact that $\mbox{dir}(\text{plane})=\mbox{...