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
149 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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1answer
777 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?
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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 ...
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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 ...
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1answer
115 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
70 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 ...
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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 ...
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0answers
108 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
89 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
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 ...
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1answer
65 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 ...
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2answers
4k 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 ...
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3answers
280 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
87 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
38 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
221 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 and B. The ...
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0answers
31 views

can an ellipse be described by two circles?

I was intrigued by the fourier visualization and the simpsons face fourier and wanted to know the answer to the question can an ellipse be described by two circles ? and will the fourier analysis ...
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0answers
251 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 ...
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3answers
97 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
133 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
71 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 ...
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0answers
83 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
36 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
90 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
41 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
303 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
111 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
91 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 ...
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1answer
183 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 ...
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2answers
218 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 ...
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1answer
190 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 ...
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0answers
45 views

Quadrature of the parabole

This exercise is from a course in mathematics history. Find U: S: V, where S is the area of ​​a parabolic segment, U is the area of the largest triangle that can fit inside the parabola segment ...
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1answer
141 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 ...
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2answers
234 views

Intersection of two tangents on a parabola proof

There are two tangent lines on a parabola $x^2$. The $x$ values of where the tangent lines intersect with the parabola are $a$ and $b$ respectively. The point where the two tangent lines intersect has ...
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3answers
91 views

parametrise equation of a hyperbola

Any point on an ellipse can be wrttien as $(a\cos\theta,b\sin\theta)$, How could we genarilse this to a hyperbola?
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0answers
30 views

find inscribed ellipses in quads

I know in an convex quads,there are a family of inscribed ellipses. what I want to konw is when the semi-axis 'a' and four vertexs are given,how to determine the rotation angle.there may be three ...
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1answer
160 views

how to simplify a general plane conic section's equation by linear algebra?

When encountering a general plane conic section a11x^2+a12xy+a22y^2+b1x+b2y+c=0, i can write it in matrix form as a quadratic form of the vector [x,y,1]. by what then? what should be done to reach the ...
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0answers
197 views

Find angle given arc length and radius

I've got a function, $r(\theta)$, of the radius of an ellipse relative to one focus of the ellipse: $$ r(\theta) = \frac{l}{1 - e\cos \theta} $$ where $e$ is the eccentricity and $l$ is the semi-latus ...
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0answers
33 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
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0answers
93 views

Locus Problem .

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola is $y^2(2x+a)=a(3x+a)^2$
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2answers
63 views

Range of a parabolic shot

Prove that the range of a parabolic shoted from a height $h$, speed $v$ and angle $\alpha$ with the floor is maximum when $\alpha$ satisfies $\cos(2\alpha) = \frac{g h}{v^2 + gh}$ This is my ...
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6answers
259 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
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1answer
59 views

Ellipse features from either expanded form or general form

I have ellipses that are not aligned with the x-axis and are not centered at the origin. Hence, their defined by either of the following two equations: ...
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1answer
92 views

Deriving Rutherford Scattering Angle expression

OK, this was an assignment and I want to be sure I am on the right track here. It's math as much as physics. We want to find an expression for $\theta$, the Rutherford scattering angle. I have an ...
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0answers
27 views

Determine the equation of the directrix of the parabola 2y=(x-1)(x-3) and find the equations of the tangents to the curve ..

Determine the equation of the directrix of the parabola 2y=(x-1)(x-3) and find the equations of the tangents to the curve at the points where the parabola cuts the x-axis Can someone please help me ...
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1answer
65 views

hyperbolic orbits, deriving in cartesian coordinates

I was working on this and I wanted to be sure I wasn't too far off. Given: $\frac{\alpha}{r} = 1 + \epsilon \cos \theta$ where $\epsilon$ is eccentricity. Also $\frac{(x + x_0)^2}{A^2} = ...
2
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1answer
64 views

Product of the distance from foci to a tangent is a constant

I am supposed to determine what is the result of said product. Given $P(x_0,y_0)$, I need to calculate the distance from the foci to the tangent line that passes through $P$, and then multiply the ...
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1answer
133 views

Kepler, cartesian coordinates and ellipses

I am trying to see if I am on the right track with this. The problem: A kepler orbit (an ellipse) in Cartesian coordinates is: $$(1−\epsilon^2)x^2 + 2\alpha \epsilon x + y^2 = \alpha^2.$$ The task ...
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1answer
262 views

Expression for hyperbola on complex plane

The hyperbola $$x^2 - y^2 = 1$$ has a simple expression in the complex plane as $\{z^2 + \bar{z}^2 = 2\}$. Is there a similarly simple expression for a hyperbola ...
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1answer
56 views

What is the vertex of this parabola and it's min value?

Th equation of the parabola is $$2\left(x+\dfrac34\right)^2−\dfrac{25}8$$ What is the vertex and the min value? and do I just plug $x$ values into the equation to get the points on the graph?