# Tagged Questions

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

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### Proof of the reflective property of the ellipese

I'm trying to prove the reflection property of the ellipses for an optics problem. The property is that that a ray of light originated at one of the ellipse's foci reflects in such a way to pass ...
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### Is this a correct way to solve this high school coordinate geometry question?

Here's the question: Given point $A$: $(-3;-1)$ Given point $B$: $(3;7)$ Given point $Z$: $(x;0)$ Find the $x$ coordinate of point $Z$ so that the angle of view of AB segment is $90$ ...
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### Finding 3rd circle's coordinate of particular radius given 2 circles coordinate, circles touch externally

Given circle say A,B,C where each of them touches each other externally . We are given radius of all 3 circles. We are also given 2-D coordinates of centre of B,C ,we need to compute coordinates of A. ...
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### Different methods for finding the minimum of $|x-2y|$ when $x^2+1=2y^2$.

For $x, y \in \Bbb R$, $x^2 + 1 = 2y^2$, find the minimum of $|x - 2y|$. At a glance I found that the point $(x, y)$ lies on a hyperbola and $|x - 2y|$ is just the distance between the point and the ...
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### Getting the coordinates of the center of a circle bisecting two other circles.

We have circles $C_1$ and $C_2$ with centers $(-d,0)$ and $(d,0)$, radii $a_1<d$ and $a_2<d$ respectively. If circle $D$ with radius $r$ (and with centre not necessarily on the x-axis) bisects ...
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### Linear functionals and hyperplanes

If $L:\Bbb R^n\to\Bbb R$ is a non-trivial linear functional , i.e $L(x+y)=L(x)+L(y), x,y \in\Bbb R^n$ and $L(ax)=aL(x), x \in\Bbb R^n, a \in\Bbb R$, then why does the set of all x $\in\Bbb R^n$ that ...
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### Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
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### the area that a part of an ellipse consumes in a square of a discrete grid

Think about a discrete grid of unit 1, which means the grid consists of infinite number of squares whose area is 1. You can assign a coordinate to each square and one of them will have the coordinate (...
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### Given any parametric curve, finding its general form?

I'll illustrate the problem I'm trying to solve with an example. Let's consider the equations $$x = \cos (t)$$ $$y = \sin (t)$$ We know that these are a parametric form of the unit circle. In ...
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### Formula for area of triangle in complex plane [closed]

If $A(z_1)$, $B(z_2)$, $C(z_3)$ are vertices of a triangle $ABC$ in Argand plane, what is the area of the triangle?
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### Plot points on an arc

I have modified this post with updated information so the problem may be more clear. Because the answer provided does not achieve the results intended, maybe adjusting the content will help adjust ...
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### Find the point on the plane xOy [closed]

Let $A(x_1; y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$ be three points not lying on the same straight line. Find the point on the plane $xOy$ such that the sum of the distances from it to these points is ...
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### Proof that if two lines are parallel then $A_1$ = $A_2$ and $B_1$ = $B_2$?

Let two lines to be parallel in their general form. $L_1$ : $A_1 x$ + $B_1 y$ + $C_1$ $L_2$ : $A_2 x$ + $B_2 y$ + $C_2$ Now i wish to prove $A_1$ = $A_2$ and $B_1$ = $B_2$ But i can only think of ...
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### Slope of axes of a General Conic Section

A General Conic Section is given by the equation $ax^2 + by^2 + 2hxy +2gx +2fy + c =0$. Let the $\theta$ be the slope of one of its axes. Prove that : $$\tan 2\theta = \dfrac{2h}{a-b}$$ ...
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### Vector Distance

let there be a line L: $\frac{x-1}{2}= \frac{y+1}{3}= \frac{z}{1}$ and a plane: $2x-y-z=5$. With this given data find: a line L1, such that L1 is parallel to L, is in P, and the distance between L and ...
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### $(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. Centroid of $\Delta ABC$ lies on $y=3x-4$, then the locus of $D$

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. If centroid of $\Delta ABC$ lies on $y=3x-4$, then what is the locus of $D$? I did try a couple of things, but I honestly ...
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### How to determine if a point lies in this particular convex region?

I have a family of hyperplanes which do not contain the origin: \begin{eqnarray} a_{11}x_1+a_{12}x_2+\dots+a_{1n}x_n &=& k_1\\ a_{21}x_1+a_{22}x_2+\dots+a_{2n}x_n &=& k_2\\ &\vdots&...
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### Check if a given coordinate lies in path of a ray (coordinate geometry)

As shown in the image I have two known coordinate pair A and B and few other known coordinate pairs (RED blob) on the graph. I need to know if any of the other given coordinates fall in line of the ...
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### Cosine Inequality

Show that given three angles $A,B,C\ge0$ with $A+B+C=2\pi$ and any positive numbers $a,b,c$ we have $$bc\cos A + ca \cos B + ab \cos C \ge -\frac {a^2+b^2+c^2}{2}$$ This problem was given in the ...
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### Where i am going wrong in finding normal to curve?

The question is Find the perpendicular distance between the normal to the curve $$x=a\cos t+at\sin t, y=a\sin t-at\cos t$$ and the origin. Equation is given in parameterized form. My attempt ...
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### Question on circles…

If three circles with radii ${3}$,${4}$,${5}$ touch each other externally at points P,Q and R,then the CIRCUMRADIUS of ∆PQR is...?? My attempt i think that the let the point of the common ...
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### Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
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### Normal vector between two parallel lines [closed]

Is there a way to calculate the normal vector of two parallel lines, without calculating the length or the points?
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### Will the boy outwit the teacher in this way? [duplicate]

In the book, Solving Mathematical Problems: A personal perspective (written by Terry Tao), he discusses a problem named (on Analytic Geometry Chapter, page 79): Problem 5.4 (Taylor 1989, p. 34, Q2)...
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### How to proove that foot of perpendicular drawn from focus to any tangent of an ellipse lie on auxillary circle?

One way is to find the foot of perpendicular and directly putting it into the equation of auxiliary circle. But that is quite a lengthy proof, is there any other short method to prove this property?
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### Locus of a point on a fixed-length segment whose endpoints slide along orthogonal lines

Suppose we have some segment $AB$ of constant length that slides in such a way that its endpoints are moving along orthogonal lines. Let $P$ be a point in the segment so that $|AP| = a$ and $|PB| = b$....
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### Can a line parallel to axis of parabola also represent tangent at a point along with the one whose slope is found using calculus?

Consider a parabola with the equation $y^2=4x$ its axis is the x-axis and vertex is (0,0) and focus at (1,0). Consider any point on the parabola say (4,4). Now we define tangent at this point as a ...
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### Partition a triangle into equal areas

A piece of wooden board in the shape of an isosceles right triangle, with sides $1$,$1$, $\sqrt{2}$ is to be sawn into two pieces. Find the length and location of the shortest straight cut which ...
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### $ax^2+by^2+2gx+2fy+2hxy+c=0$ : Understanding the equation

Given any second degree equation in $x$ and $y$, $ax^2+by^2+2gx+2fy+2hxy+c=0$ is it possible to find out the centre and/or the axis of the conic section it represents? What information can I ...
If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} -\frac{y^2}{b^2} = 1$ whose centre is $C(0,0)$ are such that $CP$ is perpendicular to $CQ$ , $a<b$ , then prove that \frac{1}{(CP)^2} ...