# Tagged Questions

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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### How to show that any rectangle in ellipse must be oriented parallel to axes?

A problem which is often given as an exercise for students learning about calculus and finding extrema, is to find maximal possible area of a rectangle inside an ellipse. Such question was asked, for ...
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### Conic Sections: Hyperbola (Finding the Locus)

This is a multipart question so bear with me until I get to the part where I am stuck on. $H$: $xy=c^2$ is a hyperbola (i) Show that $H$ can be represented by the parametric ...
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### Rotate an Ellipse

$x = h + a \cos(φ) \cos(θ) + b \sin(φ) \sin(θ)$ $y = k + b \cos(φ) \sin(θ) - a \sin(φ) \cos(θ)$ Hi, I have basic question of parametric equation for ellipse. I'm trying to rotate horizontal ellipse ...
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### Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
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### Why does the Ellipsograph/Trammel of Archimedes draw an ellipse, really?

Here's a diagram of the device I mean, hard at work drawing an ellipse. I find this quite surprising, and would like to get to the bottom of things. Essentially, a rod (black line in animation) is ...
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### 538.com's Puzzle of the Overflowing Martini Glass - How to compute the minor and major axis of an elliptical cross-section of a cone

FiveThirtyEight.com Riddler Puzzle / May 13 The puzzle goes like this; "It’s Friday. You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the ...
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### Same perimeter and area for a circle and an ellipse

For a given circle, is there exist an ellipse with same perimeter and area as to that circle? If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there ...
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### Identification of a conic section

Consider the equation $(E)\hskip 5mm x^2+xy+ky^2+6x+10=0$. I am looking for conditions on $k$ for the graph of $(E)$ to be a circle or an ellipse. Clearly, if it is a circle or an ellipse, its ...
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### 4-ellipse with distance R from four foci

I'm trying to find the equation for the generalization of an ellipse called a $n$-ellipse which has a constant distance R from four foci located at $(0,0),(0,1),(1,0),(1,1)$ Edit: As an algebraic ...
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### How do parametric equations work?

I was given a graph like this in my exam. Its defined para-metrically by x=c^2 and y =c^3. It won't help me now but could someone explain this to me why I have two seemingly different lines I know it ...
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### Slope of axes of a General Conic Section

A General Conic Section is given by the equation $ax^2 + by^2 + 2hxy +2gx +2fy + c =0$. Let the $\theta$ be the slope of one of its axes. Prove that : $$\tan 2\theta = \dfrac{2h}{a-b}$$ ...
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### Find the equation of the ellipse

An ellipse with centre at $(4,3)$ touches $x$-axis at $(0,0)$. If the slope of the major axis of ellipse is 1, then find the equation of the ellipse?
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### Find a point on an ellipse given a different point on the ellipse and either an arc length or a chord length.

Given a point $A$ on an ellipse and an arc length, how can I find the resultant point $B$ on the ellipse such that point $B$ is the arc length away from Point $A$? Alternatively, given a point $A$ on ...
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### Double Integral over the region of an ellipse cut off by a circle

I've been stuck on this question for awhile. I need to calculate the double integral $\iint_R \frac{1}{r^3} dA$ using polar coordinates. R is the region displayed below: The ellipse has centre (4,...
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### Ellipsoid Axis at density contour, why choose biggest eigen value for axis?

I've been trying to figure out how to find the density contour for a multivariate normal density function with an arbitrary number of dimensions. I've found a lot of examples for 3Dimensions and for ...
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### Find cone-plane intersection points in a construction

I have two points on the X axis, A and B, which are connected to the two points, C and D on the sketch plane parallel to XY plane. I have a point E which lies at distance h from D point in Y direction ...
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### Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter.

Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter. Here is a picture; What I have attempted; Let the parabola ...
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### Find the equation of a hyperbola, given a point on it and the length of the transverse axis

My textbook has the following question: The transverse axis of a hyperbola is of length $24$ and the curve passes through the point $(13, 10)$. Find the equation of the hyperbola. Also give the ...
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### How to get the properties of an ellipse with six points given.

I am looking for a way to calculate the lengths of both semi-axes and the rotation angle of the ellipse in the image as shown in this picture. Six points are given, with two pairs of points being ...
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### Point halfway around ellipse quadrant

I want to find the length between the centre of an ellipse and a point, P, on the ellipse, where the arc length between P and the intersection of the semi-minor axis with the ellipse is equal to the ...
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### Derive the parametric form of the locus of point where difference between distance to two points is constant

Given two points $P_1=(x_1,y_1)$ and $P_2 = (x_2,y_2)$, the locus of the point whose (signed) difference between the distance to the two points is a constant $\Delta$ is one branch of a hyperbola ...
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### Hyperbola equation proof

I've been trying to prove the canonical form of the hyperbola by myself. $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ I started from the statement that ...
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### How to derive formula for focus of a parabola?

I understand how to obtain the formula for the vertex of a formula, $y= a(x-h) + k$ where $h=-b/2a$ and the vertex is $(h,k)$. However I don't know how to get to $(h,k+1/4a)$. Could someone please ...
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### Ellipse - relation between a and b such that $F_1P \perp F_2P$

Consider the ellipse $\displaystyle \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$ with foci $F_1 (-e, 0)$ and $F_2 (e, 0)$ (where $e$ is the linear eccentricity). What is the relation between $a$ and $b$ so ...
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### How to determine standard equation of a conic from the general second degree equation?

From a given general equation of second degree i can determine the conic by following rules: Given equation: $ax^2+by^2+2hxy+2gx+2fy+c=0$ then if, $abc+2fgh-af^2-bg^2-ch^2$ is not equal to zero ...
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### Computing the properties of the 3D-projection of an ellipse.

I have an ellipse that is rotated around the white axis (see image below) in 3-dimensional space by an angle α. The axis passes through the perimeter and one of the ...
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### Ellipse from two arbitrary points, tangent at P1 and a focal point

Is it possible to find this? Really only need the semi major axis or even it's orientation. Please see the linked image. Known elements are in red and the desired element is in blue.
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### Finding the tangent of an ellipse that is perpendicular to a line

The books say's "Find the equations of the tangents to $x^2+3y^2=4$ which are perpendicular to the line $x-2y=7$" I've graphed them and found that the given line does not pass through the ...
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### circle cuts three circles at the extremities of the diameter

If the circle $$x^2 + y^2 + 2gx + 2fy + c = 0$$ cuts the three circles $$x^2 + y^2 – 5 = 0\space;\space x^2 + y^2 – 8x – 6y + 10 = 0 \space;\space x^2 + y^2 – 4x + 2y – 2 = 0;$$ at the ...
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### Algorithm: Intersection of two conics

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conic curves. The curves are given by two equations of the form:  a x^2 + b y^...
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### $(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. Centroid of $\Delta ABC$ lies on $y=3x-4$, then the locus of $D$

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. If centroid of $\Delta ABC$ lies on $y=3x-4$, then what is the locus of $D$? I did try a couple of things, but I honestly ...