Questions on conic sections and their properties; the curves formed by the intersection of a plane and a cone. Circles, ellipses, hyperbolas, and parabolas are examples of conic sections.

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6
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2answers
116 views

Given a drawing of a parabola is there any geometric construction one can make to find its focus?

This question was inspired by another one I asked myself these days Given a drawing of an ellipse is there any geometric construction we can do to find it's foci? I think this is harder, I can't ...
1
vote
1answer
22 views

Relation between a differential equation satisfied by parabolas and a formula for the slope of their tangents

Statement 1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P. Statement 2: The ...
7
votes
1answer
154 views

Ellipse inscribed on a quadrilateral

The problem is: Given that an ellipse is inscribed on a convex quadrilateral and each one of it's diagonals pass through one foci of the ellipse show that the product of the opposite sides ...
0
votes
2answers
36 views

Ellipses Conics Inquiry

A carpenter wishes to construct an elliptical table top from a sheet 4ft by 8ft plywood to make a poker table for him and his budies. He will trace out the ellipse using the "thumbtack and string" ...
0
votes
1answer
27 views

Unique Specification of Ellipse Given Two Arbitrary Axes Lengths and Axes Orientation?

Without loss of generality, let an ellipse be centered on the origin (0,0) with the major axis aligned with the 45 degree line (y=x). Given the lengths of the major and minor axes, the ellipse is ...
2
votes
1answer
76 views

Given a drawing of an ellipse is there any geometric construction we can do to find it's foci?

For example if we're given a drawing of a circle, we can take three different points on it, draw the perpendicular bisectors of them and the intersection point is the center. Is it possible to find ...
0
votes
0answers
25 views

Making Homogenous Parabola Equation

Find the locus of the mid-points of the chords of the parabola $y^2=4ax$ which subtend a right angle at the vertex of the parabola. Now we say $y^2=\frac{4ax(yk-2ax)}{k^2-2ah}$ coefficient of ...
-6
votes
2answers
144 views

Arguably the world's first differential equations

EDIT4: start of context Apologies about context. I thought that it is an all too well known reference for re-counter on the topic of differential equations. In the classical dynamic solution of ...
0
votes
1answer
15 views

Find the equation of the hyperbola that satisfies this condition

Focus is at $F\equiv(−3−3√13, 1)$, asymptotes intersect at the point with coordinates $(−3, 1)$ and one asymptote passes through $(1, 7)$ I've solved some problems that involve equations of ...
1
vote
0answers
41 views

Find the parabola given two endpoints and the midpoint along the curve

It has arbitrary orientation in 2D. I thought to equate the formulas for the arc lengths (s) between the midpoint and each end point from ...
5
votes
6answers
203 views

Area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$.

Find the area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$. First I saw that this cannot be a circle ($xy$ term), and it cannot be an ellipse with axes parallel to the coordinate axes. But ...
4
votes
2answers
60 views

Fast method to find the tangent line to a conic section: why does it work?

My teacher taught me this fast method to determine the equation of the tangent line to a conic section. In the Netherlands this is called "eerlijk delen" or literally translated into English "fair ...
-3
votes
2answers
45 views

Water Density and Fluid Force (question below) [closed]

I've been trying to study the question and the answer below. Can someone tell me how to start this problem myself? I don't understand why they named one fourth of the circle equation the whole ...
3
votes
1answer
48 views

How homogenization of line and curve works?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
2
votes
1answer
27 views

Parabola properties assumptions

I am trying to model projectile trajectory but I'm having some trouble. I didn't realise parabolas are this complicated... I have some assumptions that I would like to be clarified. If I specify a ...
3
votes
2answers
75 views

Show that $PF.PG=b^2$ in a hyperbola

If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then ...
3
votes
3answers
111 views

Property of ellipses involving normals at the endpoints of a focal chord and the midpoint of that chord

While solving a book on ellipses, I came across the following property of an ellipse which was given without proof :- If the normals be drawn at the extremities of a focal chord of an ellipse, a ...
0
votes
0answers
15 views

Measure best fitting major and minor axis length given 3 points on an ellipse

I am trying to measure the parameters of an ellipse in an image. I have the center, the rotation of the ellipse. I am trying to find the best fitting major and minor axis length based on 3 given ...
0
votes
2answers
45 views

Parabola describing projectile motion.

I am trying to create a function that will generate a parabola that describes projectile motion. Here are my inputs: The starting x-y coordinate of the throw The initial x-y velocity vector. I ...
0
votes
1answer
20 views

Test if a point is within 2 parametric “cut-off” ellipses

I have 2 parametric ellipses, both represented using the standard parametric equation of an ellipse: $$x = h + a \cos t $$ $$y = k + b \sin t $$ Lets say that the ellipses are cut-off at (see ...
0
votes
1answer
67 views

Radius vs Radius of curvature of an ellipse

I am a bit confused by the physical meaning of radius vs radius of curvature, with regards to an ellipse. For a standard ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ In this case, the $a$ ...
0
votes
1answer
24 views

Is the observed widest-width of an oblate sphere constant under all rotations?

This is something which I feel intuitively is true but I'm having trouble finding a way of proving it mathematically. Given an oblate sphere, or ellipsoid, with equation $$x^2+y^2+(z^2 / c^2)=1, ...
2
votes
4answers
106 views

Area of triangle bounded by line and degenerate “crossed lines” conic

The question is Show that the two lines given by $$(A^2 - 3B^2)x^2 + 8ABxy +(B^2 - 3A^2)y^2=0$$ and the line given by $$Ax+By+C=0$$ determine an equilateral triangle of area ...
1
vote
1answer
46 views

Half parabola- $y = x^2$ - does the derivative exist at x = 0?

If we took $y = x^2$ and cut it in half by letting $x\ge 0$, does the derivative still exist at $x = 0$ or is it $\text{DNE}$? I think it's still $0$ because the function and it's derivative are both ...
0
votes
0answers
16 views

Test if a point is within a double intersected ellipse

I have a case of 4 ellipses, every 2 ellipses represent a pipe (outer and inner), and a front and back (back being occluded by the front) My question is that is there an easy way to obtain whether ...
0
votes
2answers
33 views

Circle $x^2+y^2=2$ is stretched by a scale factor $2$ parallel to the $x$-axis, find the equation of Ellipse

What is the quick method or formula to finding this answer? Also the method for finding the answer when the stretch is parallel to the $y$-axis, Regards Tom
0
votes
1answer
56 views

Rotating an ellipse about a line

I'm attempting to solve the following question: What is the volume of the region formed when the ellipse $9x^2+4y^2=36$ is revolved around the line $2x+y=5$? My try: $$9x^2+4y^2=36$$ $$y ...
1
vote
0answers
15 views

How to find foci from size and center?

How can I calculate the foci of an ellipse, given its width, height, and center? Everything I've found uses an equation instead of parameters.
0
votes
1answer
22 views

What is the formula/method used to show that $ b=4$ in this hyperbola?

Hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ Asymptotes $y=2x$ and $y=-2x$ Also given a point $A (2, 0)$ on the hyperbola (not sure if you need this though) I have absolutely no idea how you would ...
2
votes
2answers
96 views

Area of ellipse not in xy-plane

I've got a problem in which I'm trying to find the area of an ellipse which is given by the intersection of an elliptic cylinder with a plane. Nothing here is parallel to the coordinate axes, which is ...
0
votes
1answer
51 views

Finding equation of ellipse with given point and distance between directrices

I need to find the equation of an ellipse. The given were just a point where it passes, and distance between directrices. I know that the distance between directrices is given by $2a/e$. I don't ...
0
votes
1answer
36 views

A locus problem related to conic sections

Let $(C)$ be a circle of center $O$ and radius $r$. Let $E$ be any point on $(C)$. Let $P$ be any point other than $E$ in the plane. The perpendicular bisector of $[PE]$ cuts $(OE)$ in a point $M$. ...
1
vote
1answer
34 views

Is it true that these angles are equal?

Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to ...
0
votes
1answer
32 views

Find the equation of a hyberboloid with given base, narrowest section, and the distance between them

I have one question left in an assignment and I havn't been able to solve it. I know the equaton for a hyperboloid and I know that $a$ and $b$ will be equal to each other. I don't know how to solve ...
1
vote
1answer
149 views

Equation of a parabola in 3D space

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is vertex(lowest point) of the parabola. I only know z-coordinate of this point. I need to find coordinates ...
3
votes
2answers
103 views

How do I deal with reflections inside an ellipse?

Suppose I have an ellipse with foci $F_1$ and $F_2$. How do I show that any ray of light which intersects the segment connecting the foci will have subsequent reflections that always are tangent to ...
0
votes
0answers
17 views

Length of a right triangle's hypoteneuse projected onto a sphere

Please forgive me if this is the wrong kind of question, but I need someone to verify or refute my work. One leg of a triangle has length, $b$ (base), resulting from angle theta swept out by a ray ...
-3
votes
1answer
28 views

Calculating the circumference of an ellipse

I have searched for the answer and always find that there is an open solution (integral). C1 = INTEGRAL(0->2pi) sqrt(a^2 * cos^2[t]+b^2 *sin^2[t]) dt. (a nicer page is here: ...
1
vote
1answer
70 views

Intersecting two parabolas and computing the angle between the tangents in a point of intersection

I was solving some problems on parabola. I saw a question and solved it, but my solution was way too big. The question was: If $$\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} = ...
0
votes
1answer
39 views

Equation of normal vector pointing away from ellipse

Assuming that I have an ellipse, centered at $(h,k)$ of type: $$\left(\frac {x-h}{a}\right)^2 + \left(\frac {y-k}{b}\right)^2 = 1$$ The gradient of the normal is: $$\frac{a^2(y-k)}{b^2(x-h)}$$ ...
1
vote
0answers
55 views

Proof 5 points determine a conic without projective geometry

So I'm trying to prove that any five points, of which no 3 are colinear, there is a single conic that passes through al of them. I don't want to use projective geometry but rather, only analytic ...
0
votes
1answer
68 views

Finding volume of enclosed region

The base of S is the region enclosed by the parabola $y = 9 − 9x^2$ and the X - axis. Cross-sections perpendicular to the X - axis are isosceles triangles with ...
2
votes
1answer
48 views

Geometry of the Quadratic Formula

I am well aware of proofs of the quadratic formula that show, by completing the square and other methods, that the quadratic formula is what it is. I have been scouring the Internet and other ...
0
votes
0answers
23 views

Calculate parameters of an ellipse from the analytic form

The Wikipedia provides this equation: for an ellipse, defining with a, b - half axes, x_C, y_C - Position of the middle point, Theta - Rotation. Now I need to transform this other way ...
6
votes
2answers
72 views

'3-point' curve

If you have a loop of string, a fixed point and a pencil, and stretch the string as much as possible, you draw a circle. With 2 fixed points you draw an ellipse. What do you draw with 3 fixed points?
0
votes
2answers
71 views

What is the minimum information required to define an equation for ellipse?

What is the minimum information ie. amount of points in 2-dimensional plane in order to define the equation for an ellipse? I know that unique ellipse cannot be defined when only one of the foci is ...
1
vote
1answer
92 views

Find equation of ellipse given two tangent lines at given points and a point on ellipse

I'm attempting to generate an ellipse for a stair simulation game of mine, and the inputs are: A point on the ellipse The slope of the tangent line to the ellipse at that point Another point ...
2
votes
1answer
28 views

Problem with the type of equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ and vertices?

I am asked to find the type equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ , represents ? i.e a parabola , or hyperbola or ellipse or circle by squaring twice? Now , what I have done is like this ...
1
vote
2answers
49 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
1
vote
2answers
66 views

Is it possible to find equation for ellipse when focus, eccentricity and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...