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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Proof 5 points determine a conic without projective geometry

So I'm trying to prove that any five points, of which no 3 are colinear, there is a single conic that passes through al of them. I don't want to use projective geometry but rather, only analytic ...
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1answer
89 views

Finding volume of enclosed region

The base of S is the region enclosed by the parabola $y = 9 − 9x^2$ and the X - axis. Cross-sections perpendicular to the X - axis are isosceles triangles with ...
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1answer
52 views

Geometry of the Quadratic Formula

I am well aware of proofs of the quadratic formula that show, by completing the square and other methods, that the quadratic formula is what it is. I have been scouring the Internet and other ...
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0answers
41 views

Calculate parameters of an ellipse from the analytic form

The Wikipedia provides this equation: for an ellipse, defining with a, b - half axes, x_C, y_C - Position of the middle point, Theta - Rotation. Now I need to transform this other way ...
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2answers
99 views

'3-point' curve

If you have a loop of string, a fixed point and a pencil, and stretch the string as much as possible, you draw a circle. With 2 fixed points you draw an ellipse. What do you draw with 3 fixed points?
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2answers
79 views

What is the minimum information required to define an equation for ellipse?

What is the minimum information ie. amount of points in 2-dimensional plane in order to define the equation for an ellipse? I know that unique ellipse cannot be defined when only one of the foci is ...
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1answer
172 views

Find equation of ellipse given two tangent lines at given points and a point on ellipse

I'm attempting to generate an ellipse for a stair simulation game of mine, and the inputs are: A point on the ellipse The slope of the tangent line to the ellipse at that point Another point ...
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1answer
28 views

Problem with the type of equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ and vertices?

I am asked to find the type equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ , represents ? i.e a parabola , or hyperbola or ellipse or circle by squaring twice? Now , what I have done is like this ...
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2answers
57 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
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2answers
97 views

Is it possible to find equation for ellipse when focus, eccentricity and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...
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0answers
15 views

Point on ellipse after walking a distance on the perimeter [duplicate]

I've the equation of an ellipse. Given a point (x,y) on the ellipse and a length L , I want to find the coordinates (x1,y1) of the point where I'd end up after taking a walk of length L from (x,y), ...
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1answer
110 views

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
23 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
23 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
52 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 ...
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0answers
32 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
60 views

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

A conic has eccentricity $e=0.7$, a focus $(5,−3)$ and directrix $y=2x−7$. Find the points of intersection of the conic with line $y=−3$. I'm really stuck on this, and have no idea even where to ...
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1answer
78 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
41 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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3answers
295 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
42 views

Point of intersection closest to the origin

How do I find the point of intersection of $𝑥 + 𝑦 - 𝑧 + 2 = 0$ and $𝑧^2 = 𝑥^2 + 𝑦^2$ 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
66 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 $h(x, y, z) = 0$. ...
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3answers
129 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
96 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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4answers
46 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
40 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
138 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 ...
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1answer
45 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| + ...
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1answer
241 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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3answers
56 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
65 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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1answer
131 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 + ...
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2answers
113 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. $\varepsilon$ ...
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1answer
356 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
82 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
31 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
90 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
107 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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1answer
57 views

How to find the length of the focal chord making angle $\theta$ with the axis of parabola?

A focal chord of $Y^2 = 4aX$ makes angle $\theta$ with the axis of the parabola. How can I find the length of the chord? I have used the parametric equation but couldn't go further.
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2answers
142 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 $$y−y_1 ...
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0answers
63 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 ...
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1answer
47 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: ...
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0answers
94 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
16 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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3answers
168 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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88 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
89 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 ...
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1answer
128 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
51 views

Sections of cones in higher dimensions

Everybody knows that when a plane intersects a cone at different angles and positions, we get conic sections. But, I wanted to know that if the same was possible in higher dimensions. If we take the 4 ...
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2answers
74 views

Conic sections directrix and focus

I do not understand the following: The equation of a particular parabola is: $$(y−​23)​​ = -\dfrac{1}{​16}\!\!\!\!\!​​​​(x+3)​^2​​$$ Given the equation of a parabola is - $(y−y_1)​= ...