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.

learn more… | top users | synonyms (3)

1
vote
1answer
141 views

Calculate Ellipse From Points?

How can I calculate an ellipse from a group of points ? Result: center point, x-radius, y-radius ? I'm not mathematician so I don't really know the best parameter style for ellipses. This ellipse ...
0
votes
1answer
73 views

Intersection of conics

By conic we understand a conic on the projective plane $\mathbb{P}_2=\mathbb{P}(V)$, where $V$ is $3$-dimensional. I'd like to ask how to find the number of points in the intersection of two given ...
0
votes
3answers
46 views

Find the eccentricity of the conic $4(2y-x-3)^2 -9(2x+y-1)^2=80$

Find the eccentricity of the conic $4(2y-x-3)^2 -9(2x+y-1)^2=80$ Solution : $4(2y-x-3)^2 = 4x^2-16xy+24x+16y^2-48y+36$ and $9(2x+y-1)^2 = 36x^2+36xy-36x+9y^2-18y+9$ $\therefore 4(2y-x-3)^2 ...
0
votes
4answers
163 views

Find $z$ such that $|z+1|+ |z-1|=4$

I have this problem: Find all points of the complex plane wich satisfy: $$|z+1| + |z-1| = 4 $$ I know this is an ellipse with foci 1 and -1, and i know the answer is : $$3 x^2+4 y^2 \leq 12$$ but ...
2
votes
1answer
72 views

Ellipse and circle

if $\alpha$, $\beta$, $\gamma$, $\delta$ be the eccentric angles of four points of intersection of the ellipse and any circle,prove that $\alpha+\delta+\beta+\gamma$ is an even multiple of $\pi$ ...
0
votes
1answer
88 views

Proving properties of an ellipse

I'm studying about ellipse and its properties. My reference is the following pdf: http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf My questions are from the very first page of the ...
2
votes
2answers
61 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{ ...
2
votes
1answer
90 views

How to show that this line touches the hyperbola?

The question is: $PQ$ is a chord joining the points $\phi_1$ and $\phi_2$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $\phi_1\,+\,\phi_2 = 2\alpha$, where $\alpha$ is constant, prove ...
0
votes
1answer
70 views

finding eccentricity of ellipse??

If the tangent at any point of the ellipse make an angle α with major axis and an angle β with focal radius of the point of contact then show that the eccentricity of the ellipse is given by ...
0
votes
4answers
236 views

Finding the maximum value of a function on an ellipse

Let $x$ and $y$ be real numbers such that $x^2 + 9 y^2-4 x+6 y+4=0$. Find the maximum value of $\displaystyle \frac{4x-9y}{2}$. My solution: the given function represents an ellipse. Rewriting it, we ...
1
vote
2answers
63 views

Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if ...
0
votes
1answer
505 views

Determining the major/minor axes of an ellipse from general form

I'm implementing a system that uses a least squares algorithm to fit an ellipse to a set of data points. I've successfully managed to obtain approximate locations for the centre of the ellipse but I ...
0
votes
2answers
301 views

Length of chord on ellipse

Suppose I have an ellipse centered at the origin, preferably expressed in its matrix form, and I want to know the chord length of a segment that passes through the origin with the endpoints at the ...
0
votes
2answers
557 views

Find angle at given points in Ellipse

I have Ellipse's center-points, minor-radius and major-radius. I can find, how to check if given point(x, y) exists in Ellipse or not. Now, I want to find given point(x,y) exists at which angle in ...
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
30 views

Trying to solve conic for ellipse equation

I'm trying to find out what conic the following equation represents. $9x^2+4y^2+18x-16y+24 = 0$ I know that the general ellipse equation is $(x^2)/a + (y^2)/b = 1.$ I got $9(x+1)^2 + 4(y-2)^2 = 1$, ...
1
vote
2answers
292 views

Ellipse problem : Find the slope of a common tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and a concentric circle of radius r.

Problem : Find the slope of a common tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and a concentric circle of radius r. Few concepts about Ellipse : Equation of Tangent to ellipse ...
2
votes
1answer
76 views

Coordinate System Rotation and Cross Term

If I have a conic equation $$ 5x^2 - 4xy + 8y^2 = 36 $$ and $ \left[\begin{array}{cc} 5 & -2\\ -2 & 8 \end{array}\right] $ in matrix form, whose eigenvalues are 4 and 9, how would I rotate ...
1
vote
1answer
114 views

Reflection inside an ellipse

From a typical point $P$ inside an ellipse, how many points $Q_i$ on the ellipse have $PQ_i$ normal to the ellipse? Someone asked me at school many years ago but I don't think I worked it out.
0
votes
2answers
79 views

Area of ellipse

The question is: If A represents the area of the ellipse $\,3x^2+4xy+3y^2=1$, then the value of $\frac{3\sqrt5}{\pi}A$ is For this I used rotation of axes for eliminating the $xy$ term from ...
0
votes
1answer
65 views

Conic matrix and diagonalization

If I have the conic $C$: $$ 5x^2 - 4xy + 8y^2 = 36 $$ How would I express it as a matrix of the form: $$ \begin{bmatrix}x&y\end{bmatrix} \begin{bmatrix}a&b/2\\b/2&c\end{bmatrix} ...
4
votes
2answers
82 views

Is there a latus other than the one in the rectum?

The name "Latus Rectum" sounds so very specific. Infact when I once asked why it is called as such, an explanation stated that the concave side of a parabola is called a rectum and that latus was ...
2
votes
0answers
54 views

An Easier way to solve simple equations of this type

Im currently working with ellipses and I've been given two points on a ellipse whose major axis is along the x-axis, $(4,3)$ and $(-1,4)$. The question asks me to find the length of the major and ...
1
vote
3answers
38 views

Ellipse representation

The equation $\frac{x^2}{2-a}+\frac{y^2}{a-5} +1 = 0$ represents an ellipse if $a\; \epsilon$ (A) $(2,\frac{3}{2})\;\cup\;(\frac{3}{2},5)$ (B) $(2,\frac{3}{2})$ (C) $(1,\frac{3}{2})$ (D) ...
2
votes
1answer
53 views

How large can a circle's radius be in an ellipse?

I have an ellipse centered on the origin parameterized by $a$ and $b$. Given its $x$ coordinate, how large can its radius be and still have the circle inside the ellipse?
2
votes
1answer
31 views

ellipse chord length along its axis.

how to determine the position in an ellipse, where the chord length is equal to its minor axis and perpendicular to the major axis? Is there any equation for it?
0
votes
1answer
103 views

If the segment intercepted by the parabola $y^2 =4ax$ with the line lx +my +n=0 subtends a right angle at the vertex, then

Problem : If the segment intercepted by the parabola $y^2 =4ax$ with the line lx +my +n=0 subtends a right angle at the vertex, then (a) 4al +n=0 (b)4am +n=0 (c) al +n=0 (d) 4al +4am +n=0 ...
0
votes
0answers
232 views

The vertex of the parabola is the point (a,b) and latus rectum is of length l. If the axis of the parabola is along the positive direction of y-axis

Problem : The vertex of the parabola is the point (a,b) and latus rectum is of length l. If the axis of the parabola is along the positive direction of y-axis, then its equation is (a) $(x+a)^2= ...
4
votes
1answer
137 views

Focus-Focus Definition for a Parabola

I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant. ...
1
vote
1answer
60 views

Ellipse, hyperbola and principle axis

Would anyone mind telling me how to solve (a)? I have no idea what I should do to solve this problem. Also, what is principal axes?
1
vote
1answer
255 views

How to compute the chord length of an ellipse?

How do I calculate the chord length of ellipse? I need to design a blade vane rotating inside an elliptical profile. The blade is supposed to be in contact with ellipse with a maximum clearance of ...
1
vote
2answers
61 views

Parabola : Find the points on the parabola $y^2-2y-4x=0$ whose focal length is 6 .

Problem : Find the points on the parabola $y^2-2y-4x=0$ whose focal length is 6 . Solution : The given equation $y^2-2y-4x=0$ can be written as : $ (y-1)^2=4x+1$ $\Rightarrow ...
2
votes
0answers
62 views

Best fit circle to “planetary” elliptical orbit?

I considered posting this to astronomy.stackexchange.com, but I've bugged them enough for today... Let $p(t)$ be a parametric function that traverses an ellipse such that it sweeps out equal ...
2
votes
2answers
56 views

How to find equation of a tangent

How to find equation of a tangent on a $4x^2+9y^2-24x+18y+9=0$ in $T(6,-1)$? The solution is $x=6$, but I always get: $y = \frac{-4}{15}x + \frac {3}{5} $(?!) Alternate form: This is the ellipse ...
1
vote
1answer
90 views

Concentric and Tangent Ellipse from 2 Hyperbolas

Find the equation of the ellipse that is concentric and tangent to the following hyperbolas: $$\begin{align} -2x^2 + 9y^2 - 20x - 108y + 256 &= 0 \\ x^2 - 4y^2 + 10x + 48y - 219 &= 0 ...
0
votes
1answer
81 views

Solution to a quadratic form

I'm trying to find a closed form solution of the following quadratic form for $x$. $x^{T}Dx = c$ where $c$ is just a constant placeholder for some terms on the other side. I know that, because $D$ ...
1
vote
2answers
106 views

Area of Parallelogram in an Ellipse

A parallelogram is inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with the fixed line $y=mx$ as one of its diagonals. Prove that the maximum area of the parallelogram is $2ab$. ...
1
vote
1answer
92 views

Computing a matrix to convert an (x,y) point on an ellipse to a circle

I have an ellipse defined by its semi-major axis, inclination, and position angle. The ellipse is centered on the origin. I would like to solve for a matrix that converts this ellipse to a circle. ...
1
vote
1answer
154 views

Volume of ellipsoid bounded by two planes.

I need to find the volume of ellipsoid: $$5x^2 + {y^2\over25} + {3z^2\over4} = 1$$ if the ellipsoid is bounded by $z={-1\over2}$ and $z=1$ planes. I was able to find the total volume of the ellipsoid ...
1
vote
1answer
851 views

How can convert the general form of ellipse equation in the standard form?

How can convert the general form of ellipse equation in the standard form? $$-x+2y+x^2+xy+y^2=0$$ Thank you in advance?
0
votes
1answer
56 views

Question about finding a third point on an ellipse given angle

If I have a known point $Y$ on an ellipse in the first quadrant, and known point $X$ on the $x$-axis, and some angle $\theta$ between $XY$ and $YZ$ with $Z$ being some mystery third point on the ...
0
votes
1answer
30 views

Normal to an ellipse

A normal is drawn to the ellipse $\frac{x^2}{(a^2+2a+2)^2}+\frac{y^2}{(a^2+1)^2}=1$. If maximum radius of the circle centered at the origin and touching the normal is $5$, then find the possible ...
0
votes
1answer
116 views

A problem of Tangent on Ellipse.

I have a question that requires me to find out the minimum value (length) of a segment of a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ intercepted by the coordinate axes. This is the ...
1
vote
1answer
72 views

Ellipse Problem

I have a problem which asks me to prove hat the locus of a point of intersection of the straight lines $\frac{tx}{a}-\frac{y}{b}+t=0$ and $\frac{x}{a}+\frac{ty}{b}-1=0$ where t is a parameter is an ...
2
votes
2answers
49 views

Multivariable Calculus: Volume

Trying to figure out the following problem: Evaluate the integral $\int\int\int_EzdV$, where E lies above the paraboloid $z = x^2+y^2$ and below the plane $z=6y$. Round the result to the nearest ...
3
votes
0answers
111 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
1
vote
0answers
94 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
0
votes
0answers
39 views

Is the conjugate axis in hyperbola just a number?

My maths teacher is teaching hyperbola these days, and when he drew the hyperbola, I was not able to see $b$ (in $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$) in the graph. When I asked about it, all he did ...
2
votes
1answer
68 views

Area above oblique conical section

Please pardon me if I don't use the correct terminology. Part of why I cannot solve this problem is that I don't even know what to research! Given a circle placed on top of the cone, the shape ...
4
votes
2answers
5k views

What is the focal width of a parabola?

I'm not wondering what the formula is—I already know that. For a parabola in standard form of $(x-h)^2=4p(y-k)$ I know that the focal width is $|4p|$. But what does that mean, conceptually? What ...