3
votes
0answers
51 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
1
vote
0answers
16 views

Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one. I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at ...
0
votes
0answers
75 views

Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...
2
votes
2answers
58 views

How to get radius at any specific point in ellipse

How to find radius of ellipse at any point $(x_1,y_1)$. We know semi-major axis and semi-minor axis i.e. $a$ & $b$. center of ellipse $(x_0,y_0)$. Somewhere I found. $$ r = \frac{ab}{\sqrt{ ...
1
vote
1answer
31 views

Foci Concentric Circles

My approach: Using the foci formula $$c=\sqrt{a^2-b^2}$$. By plugging in $a=3$ and $b=2$ I obtain plus and minus $\sqrt{5}$. But there's 2 choices with a root 5 result. How do i know which one is ...
1
vote
1answer
111 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
0
votes
1answer
279 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
7
votes
4answers
148 views

closest point to on $y=1/x$ to a given point

I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
1
vote
1answer
456 views

find the center of an ellipse given tangent point and angle

I have an ellipse with known major radius $r_y$ and minor radius $r_y$, aligned with the x- and y-axis. Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ ...
6
votes
0answers
83 views

What property does this equation calculate?

It's pretty difficult to Google for the meaning of a formula. This is the equation, it has to do with ellipses and GIS coordinates. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
3
votes
0answers
189 views

Ellipse and circles

Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references. Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ ...
18
votes
2answers
1k views

Intersection of two parabolae

Problem: Consider two parabolae such that their axes of symmetry form a right angle. Prove that all four points of intersection lie on a common circle (it is an assumption that there exist such four ...
11
votes
3answers
11k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
2
votes
1answer
437 views

Property of an ellipse

I need proof for the following question. Also, I want to know, can we apply the same for other conics. If yes, where and when... Please explain. Show that there exists a point K on the major axis of ...
8
votes
3answers
1k views

Minimal Ellipse Circumscribing A Right Triangle

Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one. You may chose the origin and orientation ...