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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7
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1answer
485 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\\ ...
1
vote
1answer
273 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 ...
4
votes
2answers
100 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 ...
1
vote
1answer
560 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 ...
1
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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?
2
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1answer
72 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 ...
0
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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 ...
2
votes
1answer
273 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. ...
0
votes
1answer
220 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 ...
3
votes
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 ...
1
vote
2answers
196 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 ...
1
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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!
0
votes
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
votes
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 ...
1
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0answers
49 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 ...
1
vote
1answer
900 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
168 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
210 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.
2
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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 .
0
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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
65 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
votes
1answer
190 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 ...
3
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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
votes
1answer
669 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
votes
2answers
773 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
votes
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, ...
0
votes
0answers
49 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}}\\ ...
2
votes
1answer
561 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
votes
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 ...
3
votes
1answer
653 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
votes
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
votes
3answers
78 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 $R$. ...
4
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0answers
693 views

Decomposition of a degenerate conic

As it has been done for the Intersection of conics using matrix representation the aim of this page is providing an exaustive and clear numerical example that describe the math behind the ...
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3answers
516 views

Equation of one branch of a hyperbola in general position

Given a generic expression of a conic: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0$$ is there a way to write an expression for one of the branches as a function of the coefficients? I tried using the ...
9
votes
3answers
567 views

Equal angles formed by the tangent lines to an ellipse and the lines through the foci.

Given an ellipse with foci $F_1, F_2$ and a point $P$. Let $T_1, T_2$ the points of tangency on the ellipse determined by the tangent lines through $P$. Show that $\widehat {T_1 P F_1} = \widehat {T_2 ...
2
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1answer
2k views

ellipse equation from eigenvectors and eigenvalues

I have a eigenvectors d1,d2 and eigenvalues v1,v2. The eigenvectors are axes of an ellipse that surrounds data points, with center u,v, and radii size of the eigenvalues. How can I find the ellipse ...
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0answers
133 views

Tangents to an ellipse

I was reading a section on conic sections in a book, and the author writes proofs that show that tangent lines to each of the three non-degenerate types of conic sections intersect at only one point. ...
7
votes
1answer
3k views

Intersection of conics using matrix representation

I came across a very interesting section of a wikipedia article on conics: http://en.wikipedia.org/wiki/Conic_section#Intersecting_two_conics I am trying to work out a couple of examples to add to ...
6
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2answers
22k views

Ellipse in polar coordinates

I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - Where $2a$ is the total height of the ellipse and $2b$ ...
2
votes
1answer
166 views

Help With Plugging in Values Distance Point to Ellipse

Can someone help me with plugging in the correct values in the equations given in this thread (accepted answer) -> Calculating Distance of a Point from an Ellipse Border The result values for x and ...
1
vote
1answer
59 views

Parameters of an elliptic equation?

I know that an ellipse equation described by: $\frac {x^2} {a^2} + \frac {y^2} {b^2} =1$ My question is in the equation above how many parameters we need to estimate? Two or four? The unknows ...
2
votes
1answer
227 views

How to find the equation of circle whose diameter is the latus rectum of the parabola.

The only hint given in this question is $x^2 = -36 y$ I am having problems starting the question I am clueless how to solve it.
5
votes
5answers
182 views

If $a,b \in \mathbb R$ satisfy $a^2+2ab+2b^2=7,$ then find the largest possible value of $|a-b|$

I came across the following problem that says: If $a,b \in \mathbb R$ satisfy $a^2+2ab+2b^2=7,$ then the largest possible value of $|a-b|$ is which of the following? $(1)\sqrt 7$, ...
0
votes
1answer
78 views

Perimeter of triangle inside

Given an ellipse centered at $(3,-3)$, and has a focus at $(3,-8)$. What is the perimeter of a triangle that entirely lies within the ellipse and has two of its vertices on the foci of the ellipse ...
0
votes
1answer
415 views

In an equation that looks like the standard form of an ellipse, what must the constant on the RHS equal for exactly one solution?

I am working on a homework question: What must be the value(s) of $c$ for the following equation to have exactly 1 solution? The equation is of the standard form of the equation for an ellipse, ...
7
votes
4answers
161 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 ...
0
votes
2answers
119 views

Conics generalized to surfaces of constant curvature

Do conic sections have an interesting generalization to surfaces of constant curvature? Consider a sphere (constant positive curvature) $\mathcal{S}$ centered at $O$, as well as points $A, B \in ...