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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Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
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0answers
5 views

Calculating the position of the ascending and descending nodes in an orbit

I'm working on a space sim and I'm stuck getting the position of the Ascending and Descending orbital nodes. I know how to get the position at any angle of the orbit, but it's measured from the ...
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2answers
15 views

Need help with a conic tangent question? (Hyperbolics)

I need to find the equation of the tangent to the hyperbola $$\frac{x^2}{6}-\frac{y^2}{8}=1$$ at the point $(3,2)$. I tried doing it by substituting for $y$ but the algebra is not nice at all and I ...
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5answers
21 views

How to find the tangent to a conic?

I have this question. Find the equation of the tangent to the line $y^2=x$ at the point $(16,-4)$. I have tried to use both methods to work it out. 1) Substitute $y=mx+c$ into $y^2=x$ and find a ...
2
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0answers
28 views

Extremal points relative to origin for an ellipsoid

Suppose I have an ellipsoid of the form $ax^2 + by^2 + az^2 - cxy -cyz = d$ How would I find the points nearest to, and furthest from, the origin?
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1answer
28 views

Problem involving conics. Need to find points of intersection given information about a conic.

A conic has eccentricity $e=0.7$, a focus $(5,βˆ’3)$ and directrix $y=2xβˆ’7$. Find the points of intersection of the conic with line $y=βˆ’3$. I'm really stuck on this, and have no idea even where to ...
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1answer
12 views

Hyperbola / Rotated Hyperbola Intersection

I am trying to find the point where two hyperbolas intersect, that is, to find a vertex that is common to both hyperbolas. Also, note that I am only testing for a region of both hyperbolas -- only a ...
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1answer
27 views

Help with Conic: Hyperbola's chord of contact

please help with this proof. "Show that the tangents at the endpoints of a focal chord of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ meet on the corresponding directrix." This is a ...
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0answers
35 views

What is the intersection between $x + y - z = -2$ and $z^2 = x^2 + y^2$

I got the answer as $4x + 4y + 2xy + 4 = 0$ by substituting $z = x + y + 2$ into the second equation, but I feel as this is wrong since I am missing $z$ in the function. How do I approach this ...
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2answers
45 views

Detect if two ellipses intersect

I have seen a lot of papers on how to find points of intersection between two ellipses for 2D case, but i only need to check if two ellipses are in collision. I don't need to know points of ...
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0answers
33 views

Point of intersection closest to the origin

How do I find the point of intersection of $π‘₯ + 𝑦 - 𝑧 + 2 = 0$ and $𝑧^2 = π‘₯^2 + 𝑦^2$ that is closest to the origin? I know I have to use the LaGrange multiplier in order to minimize the ...
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1answer
12 views

Finding the conic section given equations of double cone and plane

Given the function of a double cone and a plane, how do we find the intersection between them? Suppose the equation of the cone is $f(x, y, z) = 0$ and the equation of the plane is $h(x, y, z) = 0$. ...
0
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3answers
32 views

Find the point on the parabola $2y=x^{2}$ that is closest to the point $(-4,1)$

The first part of the derivative which is to the power of $-1/2$ is too small to be considered relevant. I'm not sure how to proceed from here. The answer is $(-2,2)$ but I am not sure how to get ...
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4answers
44 views

Equation of circle touching a parabola

Suppose we have a parabola $y^2=4x$ . Now, how to write equation of circle touching parabola at $(4,4)$ and passing thru focus? I know that for this parabola focus will lie at $(1,0)$ so we may ...
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1answer
30 views

Determining conic section equation given foci and sum of distance to each point

Disclaimer: This title was hard to formulate. Edits welcome. Problem: Given foci $$F_1 = (1,0)$$ $$F_2 = (3,0)$$ of a conic section, find the equation for all points $P$ that satisfy $$|PF_1| + ...
3
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4answers
39 views

Given $\vec r(t)$, what are $\vec v(t), \ v(t), \ \vec a(t), \ a(t)$?

I have come to a problem that simply states that we have a parametric curve $$\vec r(t) = (2\sin t, 3\cos t), \ \ t\in \mathbb R$$ and asks that we find $\vec v(t), \ v(t), \ \vec a(t), \ a(t)$. ...
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1answer
32 views

How to find the sum of maximum and minimum curvature in an ellipse?

I am having difficulty finding the sum of maximum and minimum curvature of the ellipse $9(x-1)^2 + y^2 = 9$. I know that I am supposed to parametrize the ellipse as $f(x(t), y(t))$, with $x(t) = 1 + ...
0
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1answer
10 views

Alternative proof of the reflection property

I'd like to prove the reflection property for the hyperbola. That is, that S'PS is bisected by the tangent at P. Suppose the tangent intersects the x axis at T. The usual method would be to use the ...
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1answer
15 views

Finding equation for diagonal ellipse given foci and eccentricity

Problem: Find the equation for the ellipse that has foci $$F_1 = (0, 0)$$ $$F_2 = (1,1)$$ and eccentricity $$\varepsilon = \frac12.$$ Hint: Use a rotation that moves the foci to the x-axis. My ...
2
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3answers
38 views

Finding the distance from ellipsoid to plane

I'm having problems with finding the distance from the ellipsoid $x^2+y^2+4z^2=4$ to the plane $x+y+z=6$. The question hinted that I'm supposed to find the distance from a point to the plane and ...
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1answer
34 views

Finding the equation for a hyperbola given foci and eccentricity

Problem: Find the equation for the hyperbola which has foci $$F_1 = (-1, 3)$$ $$F_2 = (3,3) $$ and eccentricity $$\varepsilon = 2$$ Hint: Use a translation which moves the foci to the x-axis. My ...
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1answer
386 views

Differential equation of a conic with given constraints

The differential equation of all conics whose axes coincide with the axes of coordinates is of order (A) 2 (B) 3 (C) 4 (D) 1 I know that the general equation of a conic is: If I apply ...
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3answers
49 views

Question about Ellipse [closed]

Given an ellipse, with the center at the origin and a given value of $16$ as its "$a$". There is also a point $(8,6)$ on the ellipse. The "$b$" value is less than "$a$" (so it is a wide ellipse). How ...
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1answer
456 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 ...
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0answers
41 views

Quadric and tangents planes

Let $Q$ be the quadratic $x^2 + 4xy - 2y^2 + 6z^2 + 2y +2z = 0$ Prove that $Q$ is a cone and find its vertex. Write the tangent plane $A$ to the cone in $(0,0,0)$ and say which kind of conic is the ...
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2answers
56 views

How does the eccentricity of a conic section define its shape?

Problem: Let $P$ be a point in the plane, $L$ a line containing $P$, and $\varepsilon$ a positive number. The triple $(\varepsilon, L, P)$ will then define a degenerate conic section. $\varepsilon$ ...
10
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6answers
3k views

How to find an ellipse , given 2 passing points and the tangents at them?

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ...
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1answer
41 views

How can I find the equation of a parabola only given it's x-intercepts?

I received a problem in my math class the other day that left me stumped. The problem went something like this. Mr. Lots-O-Cash would like to order a parabola that passes through the points $(-4, ...
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0answers
21 views

Finding eccentricity, directrix, foci of diagonal ellipse by rotating it

A problem I'm working on has the equation for a diagonal ellipse $$5x^2 + 5y^2 - 6xy - 8 = 0$$ which can be rotated 45 degrees to get the vertical ellipse $$\frac{x^2}{1}+\frac{y^2}{2^2} = 1$$ The ...
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2answers
26 views

Equation of tangent from a point outside it

Is there any general method of finding the equation of the tangent of a function $f(x)$ from a point $(a,b)$? $\hspace{1 mm}$ Then how do you find the angle between two tangents from $(0,0)$ to a ...
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3answers
39 views

Rotating a conic section to eliminate the $xy$ term

Problem: Given the equation $$5x^2 + 5y^2 - 6xy - 8 = 0$$ defining a non-degenerate conic section, find a rotation of the variables, such that the cross term $-6xy$ disappears in the new coordinates ...
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1answer
267 views

Determining the direct and transverse tangent lines for two non-overlapping ellipses

I am trying to determine the direct and transverse lines for two non-overlapping ellipses. I specifically mean that the two ellipses are totally separated from each other with no shared regions. I ...
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2answers
40 views

How to derive the equation of a parabola from the directrix and focus

Could someone please offer me proof and explanation of the following? - I am just having trouble with finding the '$a$' part of the equation. "The leading coefficient '$a$' in the equation $$yβˆ’y_1 ...
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1answer
23 views

Locus of the centers of the circles tangent to a given line and circle

Say you are given a circle $C$ and a straight line $l$ exterior to the circle. How to describe the set of centers of circle that are tangent to both the $C$ and $l$? I have no idea how to proceed. My ...
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2answers
50 views

Conic sections directrix and focus

I do not understand the following: The equation of a particular parabola is: $$(yβˆ’β€‹23)​​ = -\dfrac{1}{​16}\!\!\!\!\!​​​​(x+3)​^2​​$$ Given the equation of a parabola is - $(yβˆ’y_1)​= ...
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0answers
45 views

Fitting an ellipse such that the ratio of its radii is in a range

I need to fit an ellipse to a group of points. However, I have an issue and I appreciate if anyone can help me. The issue is that I need to have the fitted ellipse such that the ratio of its radii is ...
8
votes
2answers
446 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 ...
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1answer
37 views

Difficulties in understanding ellipse's minor axis's equation

I'm implementing an ellipse detector using some pdf I found on the internet, but I encounter some difficulties in understanding one of the equations. Here is the pdf: ...
7
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1answer
95 views

Group of Points on an Ellipse

I did some tooling around to find an abelian group operation for the set of points on the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1$, given by ...
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0answers
40 views

Finding equation for conic section given five points

Problem: Given the points $$(0,1),(0,-1),(2,0),(-2,0),(1,1)$$ find the equation for the conic section that passes through these points. My attempt: Using the general equation for a conic section, ...
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1answer
368 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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1answer
15 views

Reduction of general conic

The given equation is - $$3x^2 + 2xy + 3y^2 - 32y +92=0$$ To get rid of xy term i used the substitutions - $$x=p+q , y=q-p$$ Then the equation becomes - $$(p-4)^2 + 2(q-2)^2=1$$ which is an ellipse ...
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1answer
48 views

Construction of an ellipse

Is it possible to construct an ellipse with a line, compasses and a pencil? If yes, how and why is the construction correct?
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1answer
766 views

length of the focal chord

Paragraph: $PQ$ is a focal chord of the parabola: $y^2=4ax.$ The tangents to the parabola at the points $P$ and $Q$ meet at point $R$ which lies on the line $y=2x + a.$ Question: Find the length ...
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1answer
684 views

A hyperbola passing through integer lattice points

Prove that for any $n\geq 0$, there is a hyperbola that passes through exactly $n$ lattice points (= points with integer coordinates) and find an example. For example it is easy to see that the ...
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4answers
72 views

Standard Form for a Parabola

What is the standard form for the following problem? I already know that it is a horizontal parabola. I just can't seem to be able to change it into the standard format. $8yΒ² +96y-12x+240 = 0$ I ...
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0answers
26 views

Finding the equation of a rational function or a conic section given three points

I have a rational equation derived from 2 points, $(2, 2)$ and $(10, 10)$. Solving for the rational equation gives the equation $$y = \frac{20}{12-x}.$$ What I want to happen right now is that given ...
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1answer
59 views

Super conic sections?

I know graphs of the form $A x^2 + B xy + C y^2 + D x + E y + F = 0$ are conic sections. But what would happen if I changed the highest power to 3? Would this be a new 3D shape, a 4D version of it, or ...
3
votes
1answer
50 views

Why are elliptic/parabolic/hyperbolic PDEs called “elliptic”/“parabolic”/“hyperbolic”?

I see that the form of a (e.g.) parabolic equation is $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ with $B^2-4AC=0$ whereas the equation of a parabola is $$Ax^2 + 2Bxy + Cuy^2 + Dx + Ey + ...
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1answer
653 views

Calculating Intersection of an Ellipse and a Line

I found this page which gave me some equations on solving the intersection of a line with an ellipse given a point on the line and the slope of the line: There Isn't much explanation but ...