# Tagged Questions

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
28 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 ...
1answer
47 views

### Solve for an Ellipse Tangent to 2 Lines

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 ...
1answer
29 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 $(-\sqrt{13},0)$ ...
1answer
15 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 ...
2answers
44 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!
1answer
61 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 $x = 4$.
2answers
39 views

### conic sections, ellipse

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 ...
2answers
34 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 ...
2answers
33 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 ...
2answers
50 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 ...
1answer
37 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\\ ...
1answer
31 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 ...
2answers
37 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 ...
1answer
22 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 ...
3answers
54 views

### How to write this conic equation in standard form?

$$x^2+y^2-16x-20y+100=0$$ Standard form? Circle or ellipse?
1answer
20 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 ...
0answers
89 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 ...
1answer
70 views

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

In the following equation ρ(x,y) returns a constant value for a given coordinate. n is the normal vector to the surface of the form [P,Q,-1] and s is a direction vector. Using s = [Sx,Sy,Sz], the ...
0answers
28 views

### Representing conic sections as straight lines

Is there a projection that projects any conic section in a two dimensional orthogonal coordinate system with a focus at the origin into a potentially infinite set of parallel straight lines in a two ...
1answer
28 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 ...
2answers
35 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 ...
2answers
37 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 ...
1answer
51 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!
2answers
72 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 ...
1answer
43 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 ...
0answers
29 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 ...
1answer
81 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 ...
1answer
107 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!
1answer
43 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 ...
8answers
2k views

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

I was doing hyperbola ,I was thinking does it have any real life uses or it just a mathematics theory?
2answers
56 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 .
1answer
49 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 ...
1answer
100 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 ...
0answers
17 views

### Number of solutions to a conic in $Z_p$

$p\not=2$ is a prime. $a, b, c, d\in F_p$ and $acd\not =0$. C is the conic given by the homogeneous equation $ax^2+bxy+cy^2=dz^2$. If $b^2\not =4ac$, prove that #$C(F_p)=p+1$ Note: Solutions are to ...
0answers
28 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 ...
0answers
35 views

### Parabolic segment problem

I have a problem. I have tried to solve it but I get $125/6$ instead of $9/2$ (textbook result). Find the area of the parts of plane given by the solutions of the following system:  ...
1answer
56 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 ...
1answer
69 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 ...
1answer
57 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 ...
2answers
40 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 ...
0answers
18 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, ...
0answers
32 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}}\\ ...
1answer
73 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 ...
0answers
25 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 ...
1answer
57 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 ...
0answers
137 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 ...
2answers
38 views

### Write the equation of an ellipse

The information given is the focus at (-2,3), directrix y=0 and eccentricity =1/2
2answers
83 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 ...
3answers
48 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 $R$. ...
0answers
116 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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