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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2answers
109 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 ...
1
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1answer
128 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} ...
6
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2answers
109 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
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0answers
63 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 ...
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3answers
56 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
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1answer
97 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
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1answer
47 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?
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1answer
350 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 ...
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0answers
3k 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
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1answer
205 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. ...
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1answer
187 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?
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1answer
1k 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 ...
2
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2answers
112 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
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0answers
90 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
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2answers
65 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 ...
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1answer
254 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 ...
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1answer
249 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 $D$ ...
2
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2answers
307 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
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1answer
430 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. ...
3
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1answer
430 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 ...
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2answers
2k views

How to convert the general form of ellipse equation to the standard form?

How to convert the general form of ellipse equation to the standard form? $$-x+2y+x^2+xy+y^2=0$$
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1answer
69 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 ...
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1answer
58 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 ...
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1answer
299 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 ...
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1answer
105 views

Ellipse Problem

Consider a family of straight line pairs given by $\frac{tx}{a}-\frac{y}{b}+t=0$ and $\frac{x}{a}+\frac{ty}{b}-1=0$ where $t$ is a parameter. My goal is to show that the set of intersection ...
3
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2answers
60 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 ...
4
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2answers
312 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 ...
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0answers
230 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 ...
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0answers
51 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
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1answer
112 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 ...
5
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3answers
16k 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 ...
3
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3answers
762 views

Omar Khayyam's method for solving cubics

So I need to answer the following question using Khayyam's method. I can get the answer using modern methods, and I know the basics of his method, but I cannot figure out how to find the two conic ...
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1answer
120 views

Are Parabolas similar intuitively?

All parabolas are similar, but are they all similar in that it is just a question of 'zooming in and out' intuitively speaking? It seems that there should therefore be on all parabolas a curve from ...
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1answer
45 views

Why the ellipse circumference shows minor axis as 10 times?

Ellipse of having minor axis 0.692200628 and major axis 1.444667861 has circumference 6.9229....... which seems quite close to be minor axis 0.6922006.... multiplied by 10 but deviation occurs at ...
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1answer
452 views

Parabola & Area Proving (Integral)

This is not a homework question. I am a new teacher (just graduated) and a student asked me this question. The points $A(3,9)$ and $B(-2,4)$ lie on the parabola $y=x^2.$ The line $y=x+6$ joins $A$ ...
2
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0answers
585 views

Relation between ellipse general and parametric equation

I am familiar with the fact that one can relate the eigenvectors and corresponding eigenvalues of an ellipse's quadratic equation matrix, to the pose of a circle in 3-space. When say quadratic ...
0
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3answers
287 views

Proving a condition related to normal on ellipse

Prove that the straight line $lx+my+n=0$ is a normal to the ellipse $x^2/a^2 +y^2/b^2=1$ if $a^2/l^2 +b^2/n^2 = (a^2 -b^2)^2/n^2$.
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2answers
198 views

Creating an ellipse passing through a rectangle's vertices coordinates

Given a rectangle with vertices $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)$ and $D(x_4, y_4)$, how to create an ellipse with this vertices coordinates?
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1answer
88 views

Ellipse with four different radii

I need draw an ellipse in 3D (for time being, consider $z$ constant), Lets say I have center $O = (x_{0},y_{0},z_{0})$ of ellipse is at $(0,0,0)$ and radii $q_{1}, q_{2}, q_{3}, q_{4}$ of ellipse i ...
2
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0answers
205 views

Finding Quadrants of ellipse from ellipsoid of a Conic section

This is my first post here, hope I won't be giving tough time for you. I will be giving bit non relevant information here to describe my problem as it may help understand the problem better. I will ...
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0answers
96 views

Equation of ellipsoid given foci and two semi-axes

How does one find the equation of an ellipsoid given two foci, $(a,b,c)$ and $(d,e,f)$, and one semi-axis $l$? $c$ may not be equal to $f$.
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1answer
158 views

How To Simulate Mirrors/Reflection?

If light is hitting a Parabolic Trough defined by $y=x^2$ at a 60 degree angle from vertical so that the effective cross-section of the modified parabola is paramaterized by: x=t, y=t^2, z=tcot(60). ...
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0answers
63 views

How To Mathematically Slice A Parabolic Trough at an Angle?

Given a Parabolic Trough is defined as $y=x^2$ and extending infinitely in the z direction. How may I find the equation of the curve obtained through slicing the parabolic trough using a plane through ...
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2answers
610 views

Find equation of the circular cross section of a unit sphere

I have a unit sphere in Cartesian coordinates: $x^2 + y^2 + z^2 = 1$ or in spherical coordinates: $x = \rho \sin(\phi) \cos(\theta)\\ y = \rho \sin(\phi) \sin(\theta)\\ z = \rho \cos(\phi)$ I ...
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1answer
237 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
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0answers
131 views

Please help in solving $ax^2 + bxy + cx + dy + e$ = 0

Sometime back when trying to work out how to solve $ax^2 - by^2 + cx - dy + e = 0$ I learned that the way to solve such forms is to 'square the terms' and give it the form $A^2 - B^2 - E = 0$, $A = ax ...
5
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1answer
402 views

Ellipses given focus and two points

I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane. There are several possible approaches but I'm not sure which is the best - both ...
0
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2answers
433 views

Tangent to Ellipse

I've been stuck on this problem for a while now. Not quite sure how to get at it. I've tried finding the derivative of the equation and using point slope form but cant get it to look like the defined ...
0
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1answer
502 views

Implicit derivitave of a general ellipse

Consider an ellipse centered at the point $(h,k)$. Find all points $P=(x,y)$ on the ellipse for which the tangent line at $P$ is perpendicular to the line through $P$ and $(h,k)$. I know the general ...
3
votes
1answer
316 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...