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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1answer
100 views

Maximum Y in a rotate ellipse with a, b and phi

We have major axis, minor axis and the phi between major axis and y axis in a rotated ellipse. How can we find the maximum y?
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1answer
848 views

Finding eccentricity of an ellipse from latus rectum

The latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is the same as latus rectum of a parabola $y^2=4cx$ . Find eccentricity of the ellipse .
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1answer
55 views

Parabolic projectile equation demonstration question

I was looking at a book of physics and, it will sound dumb, but while I know that the maximum height equation of a projectile is max=(v·senα)/2g, I can't understand how do you get there from ...
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3answers
290 views

Would a circle overlap a parabola's bottom by more than just its vertex?

I mean, out of the condition that a circle actually crosses the parabola. My question is when a circle is "inside" a parabola, would it touch part of the parabola other than just the parabola's vertex ...
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0answers
174 views

How to find a point in an ellipse given the angle

I found a couple of formulas but I can't transform them in code. From the answer in Calculating a Point that lies on an Ellipse given an Angle , for instance, I get to: ...
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1answer
358 views

Find next point in ellipse given the chord length

I would like to draw a cloud programmatically. For this reason I need to know where to draw the next circle around the ellipse. Given the chord (circle radius), how can I calculate the next point in ...
1
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1answer
473 views

Solve for an Ellipse Tangent to 2 Lines [duplicate]

I'm trying to automate creation of a curve in PowerPoint. Here's an image of what I'm working towards: I'm trying to show a diagram of a rocket trajectory from a launch site on Earth to a circular ...
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1answer
1k views

Find equations of the ellipses given conditions on the directrices, foci, and vertices

The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation: with distance between directrices $27$, and between foci $3$; with a focus at $(-\sqrt{13},0)$ ...
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1answer
79 views

Find the equation of the hyperbola given foci and the minor axis

first time posting and using the site. I have a quick problem that I need some help with. I need to find the equation of a hyperbola given the foci and the length of the minor axis. The foci ...
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2answers
384 views

Angles and ellipse (proof)

First of all, sorry for my poor English! Can you please help me? I'm trying to prove that, given a point P at an ellipse. Please help me prove that the angles are equal. Thanks!
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1answer
1k views

hyperbola: equation for tangent lines and normal lines

Find the equations for (a) the tangent lines, and (b) the normal lines, to the hyperbola $y^2/4 - x^2/2 = 1$ when $x = 4$.
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3answers
169 views

conic sections, ellipse [closed]

A particle is travelling clockwise on the elliptical orbit given by $$\displaystyle \frac{x^2}{100} + \frac{y^2}{25} = 1$$ The particle leaves the orbit at the point $(-8, 3)$ and travels in a ...
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2answers
7k views

How do you find the distance between two points on a parabola

So I've been wanting to figure out a formula for an odd pattern I found... but to write a proof, I need to know one thing... How do I find the distance between two points on a parabola? Like, if I ...
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2answers
676 views

Finding the tangents common to two rotated ellipses?

Is there a way to find the four tangents that two rotated ellipses share? I believe that if two ellipses do not intersect and do not contain one another, they will have four tangents in common and I ...
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5answers
3k views

How to calculate ellipse sector area *from a focus*

How do you calculate the area of a sector of an ellipse when the angle of the sector is drawn from one of the focii? In other words, how to find the area swept out by the true anomaly? There are ...
7
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1answer
489 views

Determine if a conic is degenerate with the determinant.

There is a natural bijection between conics (written as homogeneous quadratic) and 3x3 matrices: $$C=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\Leftrightarrow \left(\begin{array}{ccc} a&b&d\\ ...
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1answer
275 views

Car parking problem

I want to park my car doing similar to the one in the image. But I want to define a curve such that I park the car at once (without going forward, always backward). Suppose that the place that I want ...
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2answers
101 views

How do different definitions of ellipse translate to the same thing?

There are 2 definitions of an ellipse that I know. One definition goes: The locus of a point moving in a plane such that the ratio of its distances from a fixed line (directrix) and a fixed ...
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1answer
569 views

Application of derivative - tangents to latus rectum

Drawn thru the focus of parabola is a chord perpendicular to the axis of the parabola. Two tangent lines are drawn through the points of intersection of the chord and the parabola. Prove that the ...
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3answers
6k views

How to write this conic equation in standard form?

$$x^2+y^2-16x-20y+100=0$$ Standard form? Circle or ellipse?
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1answer
74 views

How to prove and derive coefficients that an exponential whose base decreases exponentially is a parabola

Suppose I have a function defined by this recurrence-relation: $$R(0) = r$$ $$R(n) = R(n-1) * (1+G)d^{n-1}$$ Here r is a base value, G is a base growth rate (G>0) and d is a decay in the growth rate ...
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0answers
564 views

Conversion from Standard Ellipse Function to General Ellipse Function

I wonder if anyone can assist/show me how to complete this task... I have the following equation which models a dual axis magnetic field: $$\begin{equation} B_{H}^2 = B_x^2 + B_y^2 ...
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1answer
275 views

proof that intersection of two conic sections will intersect at at least two points.

In the following equation $\rho(x,y)$ returns a constant value for a given coordinate. $\mathbf n$ is the normal vector to the surface of the form $[P,Q,-1]$ and $s$ is a direction vector. ...
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1answer
221 views

Determine the Angle of an point in an Ellipse

I would like to know how to determine at which angle a point lies in an ellipse. Suppose I have an ellipse with semimajor and semiminor of 10 and 5 (see ...
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2answers
63 views

What is the rationale for the factor of $4$ in the Conics parabola equation?

The Conics form of a parabola equation is $4p(y-k)=(x-h)^2$ where $(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus. (Which is also the same distance from the ...
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2answers
203 views

Computing the Semimajor and Semiminor axis of an Ellipse

I have the equation of the ellipse which is $\frac {x^2}{4r^2}+\frac{y^2}{r^2}=1$ Putting the (4,2) point on the ellipse we get $r^2=8$ so we get the equation $\frac {x^2}{32}+\frac {y^2}8=1$ and the ...
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1answer
56 views

Different curves

I stuck on a following question. The curve is given by: $(3-k)x^{2}+(7-k)y^{2}+9x+9y+7=0$ For which parameter $k$ k the curve will present 1)ellipse or circle 2)parabola 3)hyperbola Thanks a lot!
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2answers
1k views

Finding the Width and Height of Ellipse given an a point and angle

I have ellipse, lets say that the height is half of its width and the ellipse is parallel to x axis. then the lets say the center point is situated in the origin ...
2
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1answer
1k views

Minimum distance between $x = -y^2$ and $(0,-3)$

Find the minimum distance from the parabola $x + y^2 = 0$ (i.e. $x = -y^2$) to the point $(0,-3)$. This is a homework question. When I try to use the derivative and substitute $-y^2$ for $x$, I ...
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0answers
50 views

How can I find $\int{\sqrt{\left(b^2-1\right)x^2+1\over-x^2+1}}dx$?

I got this from the perimeter of an ellipse. I came up with the formula: arclength of f(x) for x from a to b=$\int_a^b\sqrt{f'(x)^2+1}dx$. Since an ellipse has the equation: $$\left({x-h\over ...
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1answer
910 views

Incomplete elliptic integral of the second kind and the arc length of an ellipse - does a `simple` relation exist?

Short introduction For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I ...
2
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1answer
170 views

How do we know $\pi$ is a constant? [duplicate]

How did the ancient Greeks discover that the ratio of a circle's circumference to its diameter is constant? It does not seem so intuitive. Thanks!
1
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1answer
233 views

Turning an ellipse into a parabola

Today I was discussing circles, ellipses, hyperbolas, and parabolas in my precalculus class. We did the usual: completing the square, finding the center and radius (radii), etc. etc. But I like to ...
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8answers
15k views

What is the real life use of hyperbola? [closed]

The point of this question is to compile a list of applications of hyperbola because a lot of people are unknown to it and asks it frequently.
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2answers
1k views

Good books on conic section.

Can anybody suggest me good books for conics section.I want it for IIT-JEE mains and advanced and also for ISC. It should be available in India .
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1answer
66 views

Finding a,b of elipse

Given $x^{2}+y^{2}=R^{2}$, so that we multiply every $x$ by $a$ and every $y$ by $b$, $(a>b)$ And the distance between the focuses of this locus is $48R$, and the area of the rhombus which ...
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1answer
5k views

How to find equation of parabola when we only know the equation of latus rectum and coordinates of vertex?

Suppose the equation of latus rectum is x=4 and the vertex is (2,3). I am confused wouldn't there be many parabola with this same vertex and latus rectum.If not how to find the equation? The answer ...
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0answers
67 views

Equation of a general conic from 3 points and the major axis

I have read that given 3 points on a conic and the equation ($ax+by+c=0$) of its major axis, we can write the equation of the conic ($Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$). I've seen it done by ...
2
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1answer
197 views

Number of points determining a Quadric

I know that in $\mathbb{R}^2$ that 5 points in general linear position determine a unique conic (also non-degenerate). I was wondering about the higher dimensional analogue of this. Is it true, for ...
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1answer
1k views

Deriving hyperbola equation: why can we assume vertices lie in between foci?

I'm reading through a derivation of the standard equation of a horizontal hyperbola, and while I can follow the the algebra, I'm hung up on an assumption it makes early on: that the vertices lie in ...
2
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1answer
677 views

Integer solutions to a hyperbola

Is there a way to find all integer solutions to a hyperbola equation? If it helps, I am specifically looking at "square" hyperbolas (i.e. of the form $\frac{x^2}{z} - \frac{y^2}{z}=1$), where z is an ...
3
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2answers
791 views

Tangent line to a general conic at a point

If I have a conic $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ and want to know the tangent line at $(x_0, y_0)$, I thought I would just find the derivative (implicit) and use the equation of a line ...
3
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0answers
28 views

Do both these ellipses satisfy the same conditions?

I was solving a problem that asked for the equation of the ellipse with the following properties: vertex at $(-10,5)$, focus at $(-2,5)$, eccentricity $\frac{1}{2}$. I think I found two such ellipses, ...
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0answers
50 views

Representing an imperfect ellipse in 2 linear variables

I have several shapes which are roughly elliptical. I know the major and minor axes and the true circumference, so I store them like this: $$a={\text{axis}}_{\text{major}}\\ ...
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1answer
575 views

Find enpoints of major axis of an arbitrary ellipse using its general equation

I have a general equation of an ellipse in the form of $Ax^2+Bxy+Cy^2+Dx+Ey+f=0$. How do I find the equation of (or even endpoints would work) major axis of an ellipse. I am aware of following ...
4
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1answer
46 views

How does this method to find the centre work?

Say we have a conic with equation $f(x,y)=c$. My teacher says that it's centre satisfies the equations : $f_x(x,y)=f_y(x,y)=0$ (If it has a centre). She didn't give any explanation. I thought this ...
4
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1answer
689 views

Centers of the osculating circles along an ellipse

Consider an ellipse on the plane $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We will use the usual parametrization: $P(t)=(x(t),y(t))=(a\cos t,b\sin t)$. Then the tangent vector is $T(t)=(-a\sin t, b\cos ...
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0answers
2k views

Finding the eccentricity/focus/directrix of ellipses and hyperbolas under some rotation

Suppose I have an ellipse/hyperbola rotated about the origin by some angle $\theta$. Am I right in saying that the following general process will find the eccentricity $e$ of these conics? Find ...
10
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2answers
119 views

Prove that $|PF_{1}|+|PF_{2}|$ is Constant in an Elipse

Given an elipse with two focus $F_{1}$ an $F_{2}$, and $A$ is an arbitrary point at the elipse. Stright line $AF_{1}$ has another intersection point $B$ with the elipse, and $AF_{2}$ has another ...
2
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3answers
80 views

Find at least two ways to find $a, b$ and $c$ in the parabola equation

I've been fighting with this problem for some hours now, and i decided to ask the clever people on this website. The parabola with the equation $y=ax^2+bx+c$ goes through the points $P, Q$ and $R$. ...