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

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Analytically Understanding The Definite Integral As A Limit Of Sums

With naive intuition one can obviously see that the definite integral as infinite subdivisions of an area under a curve, within the finite interval "a to b", from which the function of integration ...
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19 views

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other.

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other. I tried to solve it but failed.Can someone please help me to ...
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0answers
23 views

Locus of point satisfying a condition

Consider a fixed point $O$ and $n$ fixed straight lines. Through $O$ a variable line is drawn intersecting the fixed lines in $P_1,P_2,\ldots,P_n$. On this variable line, a point $P$ is taken such ...
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1answer
9 views

Coordinate Geometry:Locus Based Problem

A rod AB of length l slides with its ends on the coordinate axes.Let O be the origin.The rectangle OAPB is completed. How to prove the locus of the foot of perpendicular drawn from P onto AB is ...
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Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$.

Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$. I have encountered this question from SL Loney.I have ...
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1answer
15 views

Intersection point [on hold]

I have line coordinate points and circle centre coordinate points and radius of the circle. I want to find the intersection point of circle and line using these coordinates and circle radius
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37 views

Equilateral triangle [on hold]

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are $(0,\,4)$ and $(0,\,0)$, find the third vertex. How many triangles are ...
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Coordinate Geometry and Straight Lines

Let A and A' be points (5,0) and (-5,0) respectively. The equation of the locus of all points P(x,y) such that ||AP|-|A'P||=8 is ?
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26 views

Straight Lines and Cordinate Geometry

The Locus of the point $P$, such that sum of squares of its distances from $(1,2)$ and $(3,4)$ is $25$ units, is $x^2+y^2-4x-6y+k=0$. Then $k =$ ?
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1answer
37 views

Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...
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20 views

Compute outer and inner outlines of graph of curves

Let's have some cubic Bezier curves and straight lines. Some of the Bezier curves and straight lines might have a shared start or end point, some might intersect. Input: A list of cubic Bezier ...
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1answer
22 views

Expression of reflection isometry in the complex plane

Using the fact that an anti-displacement in the plan has the form $$f(z) = a \overline{z} + b$$ I have done some computation to find the reflection about the line passing through two points $P$ and ...
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56 views

What theorems or frameworks explain why the roots of well-behaved functions $h : \mathbb{R} \leftarrow \mathbb{R}^2$ seem to be made up of “pieces”?

First, some terminology: given functions $g,f:Y \leftarrow X$, the equalizer of $g$ and $f$ is defined to be the set of all solutions $x \in X$ to the equation $g(x)=f(x)$ in $Y$. Okay. The following ...
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2answers
66 views

How are asymptotes actually defined in rigorous mathematics?

This question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x ...
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7 views

Request reflection matrix about these types

Supposed there's $(a,b)$ point and going to be reflected and find the mapping. The baseline formula will I use is $\begin{pmatrix} x' \\ y' \end{pmatrix}=M_{R} \begin{pmatrix} x \\ y \end{pmatrix}$. ...
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2answers
22 views

Single transformation matrix of $A \circ B$ and $B \circ A$ with certain conditions

Let $A$ is 2x1 translation matrix and $B$ is 2x2 matrix of reflection or rotation matrix (reflection, rotation, etc.). Suppose I want to find the mapping of a $y=mx+c$ line and the mapping is done by ...
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1answer
18 views

Rotation and translation of coordinate axes

I am studying rotation and translation of conical but have no doubt in basic concept (Sorry, I know this is a very stupid question but I'm really struggling to understand). Especially in this ...
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42 views

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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13 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 ${ ...
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1answer
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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
15 views

Rotate vectors around an arbitrary point on a plane

Is it possible to rotate vectors around a point on a plane? say I have some plane ax+by+cz+d=0 and a vector in form of (x0, y0, z0)+ (i, j,k)t The rotation point is in form of (x1, y1, z1) how do ...
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1answer
22 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
19 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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25 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 ...
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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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26 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
31 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
34 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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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 ...
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1answer
28 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 ...
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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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Plot Points Along plane at a distance

I have a plane $0.176776695x+0y−0.176776695z+0.35355339=0$ I select an arbitrary point $P$ on the plane. $P =(1,2,3)$ Now I want to find another point "$Q$" on the same plane that is offset $0.25$ ...
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49 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 ...
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1answer
26 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 }{} - ...
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26 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
43 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
18 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
30 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
35 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
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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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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
21 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 - ...
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29 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 ...
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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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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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25 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. ...
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27 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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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
74 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
76 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 ...