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

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Eccentricity of $9x^2 + 4y^2 - 24y + 144 = 0$

For a National Board Exam Review: Compute the eccentricity of a given curve $9x^2 + 4y^2 - 24y + 144 = 0$ Answer is $0.75$ I try: $$9x^2 + 4y^2 - 24y + 144 = 0$$ $$9x^2 + 4(y^2 - 6y + 9) = -...
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1answer
295 views

Finding the Equation of an Ellipse given the Length of the Latus Rectum and the Distance between the Foci

For a National Board Exam Review: Find the equation of the ellipse having a length of latus rectum of ${ \frac{3}{2} }$ and the distance between the foci is ${ 2\sqrt{13} }$ Answer is ${ \frac{x^...
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1answer
22 views

Converting a plane from Cartesian to Parametric

Find the equations of the following plane in both cartesian and parametric form: The plane through the point $(1,4,5)$ and perpendicular to the vector $(7,1,4)$. So far, I have obtained the ...
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1answer
32 views

Translate 2x^2 -8xy+4x+12 into the Standard form of a Hyperbola; Second Degree Term Missing

For a National Board Exam Review: What conic section is ${ 2x^2 -8xy+4x+12 }$ ? Answer is Hyperbola. But I can't seem to translate it properly to the standard form of a hyperbola.. What am I ...
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2answers
80 views

Finding the Width at the Bottom of a Vertical Parabolic Arc

For a National Board Exam Review: An arc 18m high has the form of a parabola with the axis vertical. If the width of the arc 8m from the top is 64m, Find the width of the arc at the bottom. ...
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31 views

Conic classification

I have a formula any and wonder what that is equation (hyperbola, point, lines, ellipse, parabola etc.) . However, I have doubts when I do the translation and rotation of coordinate systems. I know it'...
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2answers
1k views

What is the general equation equation for rotated ellipsoid?

I have general equation for ellipsoid not in center: $$ \frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}+\frac{(z-z_0)^2}{c^2}=1.$$ What is the equation when it's rotated based on $\alpha$(over $x$ axis), ...
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1answer
227 views

Why is a positive definite matrix needed in the ellipsoid matrix representation?

An ellipsoid centered at the origin is defined by the solutions $\mathbf{x}$ to the equation $\mathbf{x}^TM\mathbf{x} = 1$, where M is a positive definite matrix. How can I see why M needs to be ...
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2answers
37 views

$\left(a-\frac{1}{r^2}\right)\left(b-\frac{1}{r^2}\right)=h^2$

If $ax^2+2h xy+by^2=1$,prove that the maximum and minimum values of $x^2+y^2$ are given by the values of $r^2$ satisfying the relation $\left(a-\frac{1}{r^2}\right)\left(b-\frac{1}{r^2}\right)=h^2$ ...
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3answers
39 views

How to Translate two Equations for a “+/-”

For a National Board Exam Review: Find the Equation for the Asymptotes of a Hyperbola ${ (y-x)^2 - (x+5)^2 = 36 }$ Answer is ${ y-5 = \pm (x+5) }$ I've already solved the equations: here they ...
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1answer
34 views

Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle

Of the three lines $x+\sqrt3y=0,x+y=1$ and $x-\sqrt3y=0$,two are equations of two altitudes of an equilateral triangle.The centroid of the equilateral triangle is $(A)(0,0)\hspace{1cm}(B)\left(\frac{\...
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1answer
36 views

Prove that distance of $P$ from either of the points of contact is $\sqrt{\frac{abc}{a+b+c}}$

Three circles of radii $a,b,c$ touch one another externally and the tangents at their points of contact meet at a point $P$.Prove that distance of $P$ from either of the points of contact is $\sqrt{\...
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8answers
3k views

Prove the theorem on analytic geometry in the picture.

I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in ...
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2answers
76 views

Determine the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ such that it has the least area but contains the circle $(x-1)^2+y^2=1$

Determine the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ such that it has the least area but contains the circle $(x-1)^2+y^2=1$ Since the area of ellipse is $A=\pi ab\Rightarrow A^...
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1answer
67 views

Distance of the Focus of an Hyperbola to the X-Axis

For a National Board Exam Review: How far from the $x$-axis is the focus of the hyperbola $x^2 -2y^2 + 4x + 4y + 4$? Answer is $2.73$ Simplify into Standard Form: $$ \frac{ (y-1)^2 }{} - \frac{...
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0answers
55 views

Easy question, hard solution: find the area about a domain in the plane?

We want to find the area of a domain with piecewisely smooth boundary by using the coordinates $(p,\theta)$ of the random line: It has been known that every straight line $\ell$ on $R^2$ can be ...
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1answer
73 views

Prove that the line $PQ$ passes through a fixed point

A right isosceles triangle $AOB$ ($O$ being the origin), is such that when $AO$ and $BO$ are extended to points $P$ and $Q$ the relation $2AP.BQ=AB^2$ holds. Prove that the line $PQ$ passes through a ...
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1answer
212 views

Parabolic Cable Suspended; Inconsistent Latus Rectum and Equation of Line

For a National Board Exam Review: A cable suspended form supports that are the same height and 600ft apart has a sag of 100ft. If the cable hangs in the form of a parabola, find its equation ...
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4answers
50 views

Tangent to the x-1 Axis

For a National Board Exam Review: Point (3,4) is the center of the circle tangent to the x-1 axis. What is the point of tangency? Answer is (3,0) I usually would provide an attempt but I do ...
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2answers
44 views

Prove that the line $CQ$ passes through a fixed a point

Given $A(3,0)$ and $B(6,0)$ are $2$ fixed points and $P(x,y)$ is a variable point. $AP$ and $BP$ meet the y axis at $C$ and $D$ respectively. The line $OP$, $O$ being the origin intersects the line $...
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1answer
39 views

Find circumradius of $\Delta DEC$

$A(0,0),B(4,0)$ and $C(5,-2\sqrt 6)$ are the vertices of $\Delta ABC$. Incircle of the triangle touches side $AC$ and $BC$ at $D$ and $E$ respectively. Find the circumradius of the triangle $DEC$. Is ...
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2answers
33 views

Intuitive Way to calculate Volume of the Solid bounded by a Plane

For a National Board Exam Review: What is the volume of the solid bounded by the plane $3x+4y+6z=12$ and the coordinate axes? Answer is $4$. I am looking for a quick and intuitive way to ...
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2answers
327 views

Finding the other end of the Diameter

For a National Board Exam Review A circle has it center at (3, -2) and one end of a diameter at (7,2). Find the other end of the diameter. Answer is (-1,6) $${ m = \frac{ y^2 - y^1 }{ x^2 - x^...
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1answer
36 views

Problem with a General Equation of $2^{\mathrm{nd}}$ degree

Problem: If the lines joining the origin and the points of intersection of curves $ax^2+2hxy+by^2+2gx=0$ and $a_1x^2+2h_1xy+b_1y^2+2g_1x=0$ are mutually perpendicular, prove that $g(a_1+b_1)=g_1(...
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4answers
81 views

Please help me with this problem on Family of Lines

Problem: Consider a family of straight lines $(x+y)+\lambda(2x-y+1)=0$. Find the equation of the straight line belonging to this family which is farthest from $(1,-3)$. $$$$ Unfortunately I've ...
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1answer
38 views

Ellipse or hyperbola?

$C$ is the equation $$-2x^2+6xy+6y^2 = 1.$$ How can you see whether it is an ellipse or a hyperbola? I've calculated the eigenvalues and eigenvectors but I don't know how to continue. Thanks!
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3answers
90 views

Shortcut for Finding the Equation of a Line as a Median of a Triangle

For a National Board Exam: The points A(1,0), B(9,2), C(3,6) are vertices of a triangle. Which of the following is an equation of one of the medians? Choices are: A. ${7x-y=23}$ B. ${x-7y=23}...
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2answers
48 views

Intercept made by a line between two concentric circles

Let $$x^2+y^2-9=4r^2\enspace (r=1,2,3)$$ be $3$ concentric cirlces. Prove that the intercept made by line $$3x+4y+15=0$$ between any two cirlces is same. I thought of calculating the intercept ...
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2answers
184 views

Please help me solve this problem on Coordinate Geometry

Problem: A rod of length 2 units moves so that its ends are on the positive X-axis and on the line $x+y=0$ which lies in the second quadrant. Find the locus of the midpoint of the rod. I've ...
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5answers
120 views

Perpendicular Bisector of Made from Two Points

For a National Board Exam Review: Find the equation of the perpendicular bisector of the line joining (4,0) and (-6, -3) Answer is 20x + 6y + 29 = 0 I dont know where I went wrong. This is ...
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1answer
136 views

This is an ISI B.Math Entrance Test Problem

How should I find the difference between the radii of the smallest and the largest circles, which have their centres on the circumference of the circle $x^2+2x +y^2+4y=4$ and pass through the point $(...
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2answers
37 views

How $f_1(x,y,z)=c_1$ and $f_2(x,y,z)=c_2$ result in a curve in $\mathbb R^3$?

Though a usual way to represent a curve or an arc in space is by particularization, there is also another way to define a curve in space. Introduction: A general curve in $\mathbb R^2$ can be ...
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4answers
9k views

False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I ...
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1answer
35 views

How to find a transversal of two lines that is also perpendicular to a plane

I have this problem where I have two lines given and I have to find a transversal. However, it also has to be perpendicular to a given plane (lines are not necessarily in the given plane). My guess ...
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1answer
171 views

Does a tangent exist at $x=0$ to $y=sgn(x)$?

Yesterday my professor told me that a tangent can be constructed at $x=0$ to the signum function reasoning that the two points considered while drawing a tangent must be close horizontally and not ...
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1answer
109 views

Calculate point of intersection line of two planes

I found some source code that I do not really understand. I will give some pseudo-code in my description to give you a better idea how the algorithm works. Basically, two planes with three vertices ...
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0answers
82 views

. Find the projection of the triangle on the coordinate planes.

Given the following, three vectors: a⃗ =3i−2j+5k b⃗ =i−6j+6k c⃗ =2i+3j−k Relative to cartesian coordinate systems with origin O. I calculated the sides to be 4.58,11.45 and 7.87. I also calculated ...
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1answer
75 views

Equation of parabola from 2 points and an angle at the first point

A parabola starts at a coordinate A and ends at coordinate B. Angel of tangent through A is given 'theta'. With these data (A,B,theta) how can I get an equation of a parabola?
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General formula or at least check of existence of some given n-1 polytope cross section obtain by one cut on a given n polytope?

This is a curious observation inspired after beating this game So for these two levels in question, after some prior experience in common cross sections and loads of trial and error, I found it is ...
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2answers
88 views

Values of $k$ for which the line $y=kx-1$ is tangent to the parabola with the equation $y=x^{2}+3$

How can I find the values of $k$ for which the straight line $y=kx-1$ is tangent to the parabola with the equation $y=x^{2}+3$? I used this short cut form $c=-am^{2}$, which gives me $k=\pm 2$. I ...
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1answer
61 views

How to find the length of a line segment inside of the unit cube?

$$\frac{x}2=y=\frac{z}{-2}$$ What is the length of the segment inside of the unit cube? I guess I should find the intersections of the line and the $x=1,y=1,z=1$ planes but I think this line doesn'...
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3answers
104 views

Orthocenter of triangle $DEF$ is same as the circumcenter of triangle $ABC$

$D,E,F$ are mid points of the sides of the triangle $ABC$,then prove that the orthocenter of triangle DEF is same as the circumcenter of triangle ABC. I cannnot figure out what coordinates to suppose ...
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0answers
40 views

$1 \leq a^2x^2 + b^2y^2 - abxy \leq 9 , x\geq 1$- question compactness and connectedness..

I was told that this object was a cone, I cannot see that, can anyone tell me how to identify which object this is, so as to continue assesing and answering questions of compactness and connectedness.....
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2answers
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Applications of derivatives Analytical geometry

Any tangent at a point $P (x,y)$ to the ellipse $x^2/8 + y^2/18 =1$ meets the coordinate axes in the points $A$ and $B$ such that the the area of triangle $OAB$ is least where $O$ is the origin. Then ...
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1answer
19 views

If the tangents at

If the tangents at $P(1,1)$ on the curve $y^2 =x(2-x)^2$ meets the curve again at $Q$ then points of $Q$ is of the form $(3a/b,\, a/2b)$ so I have to find $a$ and $b$.
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2answers
141 views

Points of intersection of two lines

I have two lines that are concurrents and I want to know the point of intersection between them. To find the point my algorithm performs the following equation and replacing the lambda found in one of ...
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1answer
29 views

Coincidents lines

My algorithm to determine whether two lines are coincident (which have been proven previously they are parallel) verifies the following equation: $$ \dfrac{x - x_o}{a} = \dfrac{y - y_o}{b} = \dfrac{z ...
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1answer
91 views

Intersection point of two lines in 3D

I need an algorithm that returns the point of intersection between two lines. The algorithm is capable of determining the relative position then I'm sure the lines will intersect. My question is: I ...
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4answers
187 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
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1answer
33 views

Equation of BI,CI given and angle A to be found

If $I(1,0)$ is the center of incircle of triangle ABC,the equation of BI is $x-1=0$ and the equation of CI is $x-y-1=0$,then angle BAC is (A)$\frac{\pi}{4}$ (B)$\frac{\pi}{3}$ (C)$\frac{\pi}{2}$ (...