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

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2
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2answers
43 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
6 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
46 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
vote
1answer
47 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
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1answer
34 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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vote
6answers
75 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
66 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
311 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
34 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
votes
1answer
26 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
votes
2answers
28 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
53 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
46 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 ...
1
vote
3answers
44 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 $2x+y+6=0$?
2
votes
2answers
43 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
19 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
62 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
30 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
38 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) ...
0
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2answers
23 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
2answers
51 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
26 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
votes
2answers
79 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
34 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 ...
-1
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1answer
30 views

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 ...
1
vote
2answers
42 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
votes
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
17 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
55 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
13 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
0
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1answer
35 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 ...
1
vote
0answers
35 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
23 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
43 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
20 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
20 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 ...
-5
votes
1answer
37 views

Problem based on Equation of straight lines. [closed]

Find the equations of a pair of straight lines which pass through origin and perpendicular to each lines represented by $ax^2+2hxy+by^2=0$. I have already posted this question few days before but ...
0
votes
1answer
34 views

Equation of parabola whose ends of latus rectum are $(-1,2)$ and $(5,2)$

I found the distance between ends using distance formula i.e $6$. $\Rightarrow 4a => $ $a= 3/2$ and the focus $(2,2) $ What should I do next? How to use this information in $(x-h)^2 = -4a(y-k)^2$
0
votes
1answer
30 views

Finding shortest distance from a point to line through direction vector

Find the shortest distance from a point $P(1,3,-2)$ to the line through $P_0 (2,0,-1)$ with direction vector $d = (1, -1, 0)$. I know how to find distance between a point $(x,y)$ and a line ...
0
votes
2answers
94 views

The Locus of the Centroid of a Variable Triangle [closed]

For the variable Triangle $\Delta ABC$ with fixed vertex at $C(1,2)$ and $A,\,B$ having co-ordinates $(\cos t, \sin t)$, $(\sin t, -\cos t)$, find the locus of its centroid. I have already asked this ...
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votes
1answer
19 views

Equation of a Pair of Straight Lines [2nd degree] [closed]

If $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of straight lines then show that the square of the distance from origin to their point of intersection is $\cfrac{c(a+b)-f^2-g^2}{ab-h^2}$ I could ...
2
votes
1answer
39 views

Centroid of a Triangle and Cosine Law

In $\triangle ABC$, $M$ and $N$ are midpoints of $BC$ and $CA$ respectively such that $AM=14$ and $BN=8$. If $\angle C= 60^{\circ}$, find the length of $AB$. For simplicity sake, let $x=AB$, ...
6
votes
1answer
126 views

Derivative of intersection volume

Let $K$ be a convex body in $\mathbb{R}^n$ and set $f:\textrm{SL}(n)\rightarrow \mathbb{R}$ as $f(T)=\textrm{Vol}_n (TB\cap K)$ where $B$ is the Euclidean unit ball. How can we find extreme points of ...
1
vote
0answers
46 views

How to find the tangent cone to a set in a point?

Let $S\in R^{n}$ is a set and $x\in S$. We define tangent cone of $S$ in $x$ as: $$T_{S}(x)=\{z\in R^{n}:\exists (x_{k}), x_{k}\in S, x_{k}\rightarrow x, \exists (y_{k}), y_{k}>0, ...
1
vote
1answer
34 views

Homogeneous Equation of second degree in x and y.

Find the single equation of the two lines through the origin and perpendicular to the each lines represented by $ax^2+2hxy+by^2=0$ I tried the factorization of the given equation but it was fail..
0
votes
1answer
39 views

Equation of a Straight lines.

A variable straight line drawn through the point of intersection of the straight lines $\frac xa + \frac yb=1$ and $\frac xb + \frac ya=1$ meets the co ordinate axes at $A$ and $B$. Prove that the ...
1
vote
1answer
32 views

Finding distance of points of intersection of curve with another form its center.

Let $C$ be a curve which is locus of point of intersection of the lines $x=2+m$and $my=4-m$. A circle $S:(x-2)^2+(y+1)^2=25 $ intersects the curve $C$ at four points $P,Q,R,S$. If $O$ is the centre of ...
1
vote
1answer
44 views

Finding equation of circle knowing 1 point, radius and that it touches the x-axis

I'm currently studying circle co-ordinate geometry, and this problem has puzzled me. Find the equations of the circles of radius $5$, which touch the x-axis, and pass through the point $(3,1)$. I ...