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

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4
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1answer
60 views

Triangle with same black and white areas

Suppose we have an infinite chessboard with the usual black/white coloring. A triangle $T$ with area $a$ is given with vertices at corners of some cells. Prove that there exists another triangle $T'$ ...
0
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1answer
30 views

Locus of vertex

A variable parabola of latus rectum $l $, touches a fixed equal parabola , the axes of the two curves being parallel . Then locus of the vertex of the moving curve is a parabola , then what is the ...
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2answers
56 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola [closed]

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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0answers
42 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
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2answers
50 views

Prob. 19, Chap. 1 in Baby Rudin: For what $\mathbf{c}$ and $r > 0$ does this equivalence hold?

Here's Prob. 19 in Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $\mathbf{a} \in \mathbb{R}^k$, $\mathbf{b} \in \mathbb{R}^k$. Find $\mathbf{c} \in ...
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0answers
33 views

Prob. 16, Chap. 1 in Baby Rudin

Here is Prob. 16, Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $k \geq 3$, $\mathbf{x}, \mathbf{y} \in \mathbb{R}^k$, $\vert \mathbf{x} - ...
0
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1answer
49 views

Proof of equal angles in a quadrilateral.

points E and F are given on side BC of a convex quadrilateral ABCD (with E closer than F to B). Suppose angle EAB = angle CDF and angle FAE = angle FDE. Prove that angle CAF = angle EDB.
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4answers
103 views

Equation of a circle tangent to two lines , given the radius . [closed]

What is the equation of the circle whose center is in the first quadrant and with the radius of $4$ units, given that it is tangent to the $x$-axis and to the line $4x-3y=0$?
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2answers
50 views

How to calculate one of the vectors that generate a given cross-product?

Given the vector: $$\vec b=(-0.361728, 0.116631, 0.924960)$$ and it's cross-product: $$\vec a \times \vec b=(-0.877913, 0.291252, -0.380054)$$ How do I calculate $\vec a$ ? It's been a while since ...
1
vote
3answers
33 views

Reflecting coordinates over the line $x = -1$

I know how to reflect a coordinate over the $y$ and $x$ axis, but is there a rule I could use to help me find the reflected point over $x = -1$? This is what I know already: Over the $x$-axis: ...
1
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2answers
41 views

General form of intersection line between 2D plane and 3D hyperboloid when offset from symmetry axis?

General Background Suppose I have one side of a two-sheet hyperboloid as a general three dimensional shape, where the symmetry axis is along, say, my x-axis ($\hat{\mathbf{x}}$) in my chosen ...
3
votes
1answer
37 views

remove the interior points of two intersected closed-curves

The problem is as follows I have two intersected closed curves and each curve was represented by two arrays respectively, which means we know the coordinates of every points $(x_i,y_i)$ but no ...
2
votes
2answers
54 views

How can one measure distance between point and the line in maximum metric space?

Given metric space $M = (\mathbb{R}^2, d)$ where $d = \operatorname{max}\{|x_1 - y_1|, |x_2 - y_2|\}$, how can one measure distance from some arbitrary point $X$ to the line $y = 3$, let's say? How ...
0
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1answer
9 views

Finding the equation of a coaxial circle with its diameter falls on the radical line

Here is the problem:- $L: x – y + 3 = 0$ is the radical line for $S$, the system of coaxial circles. $C: x^2 + y^2 – 2x – 4y – 11 = 0$ is a member of $S$ with $AB$ as the common chord. (a) Find the ...
0
votes
2answers
56 views

Tangent line of Lissajous curve?

I'm trying to find at how many points the tangent line of $(\cos(3t),\sin(2t))$ goes through the point $(3,0)$. My attempt: This is the same thing as saying for how many values of $t$ do we have ...
1
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1answer
64 views

The tangent at a point $P$ on the curve $y=\ln(\frac{2+\sqrt{4-x^2}}{2-\sqrt{4-x^2}})-\sqrt{4-x^2}$ meets the $y-$axis at $T,$then find $PT^2.$

The tangent at a point $P$ on the curve $y=\ln(\frac{2+\sqrt{4-x^2}}{2-\sqrt{4-x^2}})-\sqrt{4-x^2}$ meets the $y-$axis at $T,$then find $PT^2.$ Let the point of tangency be $P(x_0,y_0)$ on the ...
0
votes
1answer
26 views

Find the equation of a plane from a line

Let $L$ be a line that passes through points $a = (1,-1,-2)$ and $b =(2,-1,1)$. Let $V_1$ be the plane $x+y-3z+6=0$. Find the equation for $L$. Find the equation for the plane $V_2$ that ...
0
votes
1answer
35 views

Area of the triangle ABC is $\frac{r^5}{2fgh}$

Through a point P(f,g,h) a plane is drawn at right angles to OP where 'O' is the origin, to meet the coordinate axes in A,B,C.Prove that the area of the triangle ABC is $\frac{r^5}{2fgh}$ where OP=r. ...
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6answers
78 views

Understanding what a plane is in $\mathbb R^3$

I understand how spheres circles and so on work, My interpretation comes from the sum of their co-ordinates equals the $radius^2$. I understand how this works but with planes Im really confused. The ...
2
votes
1answer
83 views

A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find $P$ and $Q$.

A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find the coordinates of $P$ and $Q$. I found the lengths of the lines $AQ$ and ...
4
votes
4answers
329 views

Can you find the treasure??

My big bro gave this problem one week ago. I could not still solve it.Please HELP. STORY A man was just looking for items in his store room. Suddenly he found a map , which showed then it stated ...
0
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2answers
81 views

finding the radius of the circle given a coordinate

find the radius of the circle with center at (-1,2) if a chord of length 10 is bisected at (4,-3).(this is exactly what our professor given to us) im thinking of using the distance formula which is ...
0
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1answer
31 views

Finding vertices of rhombus formed by lines $y=2x+4$, $y=-\frac{1}{3}x+4$ and $(12,0)$ is a vertex. Can't find last vertex.

The equations of two adjacent sides of a rhombus are $y=2x+4$, $y=-\frac{1}{3}x+4$. If $(12,0)$ is one vertex and all vertices have positive coordinates, find the coordinates of the other three ...
0
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2answers
30 views

Maximum area of $\Delta QSR$

The circle $C \equiv x^2+y^2=1$ cuts $X$ and $Y$ axes at $P$ and $Q$ Respectively. if another circle with centre $Q$ and variable radius is drawn so that it meets $C$ at $R$ and the line $PQ$ at $S$. ...
0
votes
2answers
73 views

Equation of a cone

Find the equation of the cone whose vertex is at the origin and whose directing curve is given by the equations: $$\begin{cases} x^2-2z+1=0 \\ y-z+1=0\end{cases} $$ We know that an eliptic cone is ...
0
votes
1answer
59 views

A circle inscribed in a rhombus.

A circle is inscribed (i.e. touches all four sides) into rhombus ABCD with one angle 60 degree. The distance from centre of circle to the nearest vertex is 1. If P is any point on the circle, then ...
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3answers
54 views

Find the center of circle given two tangent lines (the lines are parallel) and a point.

How to find the center of a circle if the circle is passing through $(-1,6)$ and tangent to the lines $x-2y+8=0$ and $2x+y+6=0$?
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votes
2answers
57 views

If the equation of side BC is $2x-y=10$,then find the possible coordinates of vertex A.

Let ABC be a triangle having orthocentre & circumcentre at $(9,5)$ and $(0,0)$ respectively. If the equation of side BC is $2x-y=10$,then find the possible coordinates of vertex A. MY ...
0
votes
2answers
21 views

Find the locus of the point R on L such that the distances BP,BR and BQ are in harmonic progression.

A variable line L passing through the point $B(2,5)$ intersect the lines $2x^2-5xy+2y^2=0$ at P and Q.Find the locus of the point R on L such that the distances BP,BR and BQ are in harmonic ...
0
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4answers
67 views

Is such a triangle possible?

a triangle angle bac is 45 degrees . side bc is 4 units . altitude from point a is 4 units. Apart from a right angle triangle where altitude becomes side ac is another such triangle possible??
1
vote
1answer
32 views

Area bounded by Point $P$ in xy plane, If $\max\left\{\bf{PA+PB\;,PB+PC}\right\}\leq 2,$

A point $P$ moves in $xy$ plane such that $\max\left\{\bf{PA+PB\;,PB+PC}\right\}\leq 2,$ Then Area of the Regine Bounded by Point $\bf{P}$ is, If Coordinate of $A(0,0)\;\;,B(1,0)$ and ...
0
votes
1answer
64 views

How to determine general form of line equation in 3D from 2 points without using vectors, matrices, etc

For a 2D line equation in General Form ($ax + by + c = 0$) it is possible to calculate all coefficients from two given points as follows: $a = y_1-y_2$ $b = x_2-x_1$ $c = (x_1-x_2) y_1 + (y_2-y_1) ...
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votes
2answers
27 views

Locus of point of intersection of tangents at $A$ and $B$

From a Point $P$ on $C_1 \equiv x^2+y^2=9$ two tangents are drawn to $C_2 \equiv x^2+y^2=1$ which meets $C_1$ at $A$ and $B$. Find the Locus of point of intersection of tangents at $A$ and $B$ on ...
2
votes
3answers
66 views

Prove the median of right triangle is half the length of the hypotenuse

How can I prove that in a right triangle the median which tends to the hypotenuse has length half of that the hypotenuse? I want to show using vector methods or analytic geometry
1
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1answer
29 views

Help With Steps Of Finding Orthocenter

I'm trying to find the orthocenter of $M(-8,0)$, $N(0,0)$, $P(-4,6)$. I thought I did all of the steps right but I keep getting an answer of $(-4,6)$, but my book says $(-4,2.6667)$. Here are the ...
0
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2answers
84 views

Showing two lines on a triangle coincide

Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$. Suppose that the altitude drawn from $A$ intersects the circle at $N$. Draw two lines through circumcenter $O$ of ...
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2answers
68 views

Finding the angle b/w two lines in Coordinate Geometry

In my coaching class I was taught that the tangent of the angle between two lines having slopes $m_1$ and $m_2$ is given by the formula modulus of $\frac{m_1-m_2}{1+m_1m_2}$. We can then use ...
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1answer
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Proof Problem on Homogeneous Equation Of Second Degree

If the lines represented by the equation $x^2 + y^2= c^2\left(\dfrac{bx+ay}{ab}\right)^2 $ form a right angle, prove that: $$\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}=\frac{3}{c^2}$$ I don't ...
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2answers
50 views

Parabola having focus $(1,2)$ touches both axes. Find the equation of directrix.

Parabola having focus $(1,2)$ touches both axes. Find the equation of directrix. As perpendicular tangents meet at directrix, the directrix passes through origin. So the directrix has equation of the ...
0
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2answers
42 views

Showing boundedness of a set defined by equality

I am trying to show that the set given by: $$S = \{\mathbf x \in \Bbb R^2 \mid x^2 + 3xy + 3y^2 = 3\}$$ is bounded. I am able to show that this is true whenever $(x,y) \in \Bbb R^2$ is such that: ...
0
votes
1answer
18 views

Formally showing that there exist exactly four isometries of $\mathbb{E}^2$ that map two intersecting lines

Given are two intersecting lines $l$ and $l'$ in $\mathbb{E}^2$. How does one show that there are exactly four isometries that map $l$ to $l'$ and have $l\cap l'$ as fixed point? Intuitively, I've ...
2
votes
2answers
60 views

Where to shoot to hit a moving target in 3D space

Typical problem of many computer games and also reality (targeting computers of modern jets or anti-aircraft systems): You have a target at known position and it is flying at known constant velocity. ...
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0answers
19 views

Chord of one ellipse tangent to other

After finding equation of PQ I tried putting value of y from PQ in other ellipse and then set discriminant=0. But it is getting too tedious
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1answer
55 views

Tangent plane of a surface

Find the equation of the tangent plane of of the following surface patch at the indicated point: $$ σ(r, θ) = (r \cosh θ, r \sinh θ, r^2), (1, 0, 1).$$ I know that the tangent space of a surface ...
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vote
0answers
42 views

Pythagorean Theorem via Geometric Progression

Cut The Kont offers a proof of the Pythagorean Theorem based on a converging geometric series of similar right triangles. The second image on that page (linked) is the most relevant for this question. ...
0
votes
1answer
27 views

Find the length of latus rectum of the conic $7x^2+12xy-2y^2-2x+4y-7=0$.

Find the length of latus rectum of the conic $7x^2+12xy-2y^2-2x+4y-7=0$. The given conic $7x^2+12xy-2y^2-2x+4y-7=0$ is a hyperbola because when i compare it with $ax^2+2hxy+by^2+2gx+2hy+c=0$ and ...
0
votes
0answers
22 views

Length of a focal chord [duplicate]

how to prove that that length of focal chord of standard ellipse(a>b) which inclined angle titha to the major axis is 2ab^2/(a^2sin^2θ+b^2cos^2θ I tried Equation of focal chord inclined at angle ...
0
votes
1answer
50 views

Find the eccentricity of the ellipse $(x-3)^2+(y-4)^2=\frac{y^2}{9}$

Find the eccentricity of the ellipse $(x-3)^2+(y-4)^2=\frac{y^2}{9}$ $(x-3)^2+(y-4)^2=\frac{y^2}{9}$ $x^2-6x+9+y^2-8y+16-\frac{y^2}{9}=0$ $(x-3)^2+\frac{8y^2}{9}-8y+16=0$ ...
1
vote
0answers
21 views

Complex geometry intersection of lines

Let $A,B,C,D$ be points. Prove that $AB\cap CD=\frac{(\overline{a}b-a\overline{b})-(a-b)(\overline{c}d-c\overline{d})}{(\overline{a}-\overline{b})(c-d)-(a-b)(\overline{c}-\overline{d})}$. (Here the ...
0
votes
1answer
29 views

A point $P(a,b)$ is equidistant from the y-axis and from the point $(4,0)$. Find a relationship between $a$ and $b$.

A point $P(a,b)$ is equidistant from the y-axis and from the point $(4,0)$. Find a relationship between $a$ and $b$. I know that the distance of $(a,b)$ from the point $(4,0)$ is $\sqrt ...