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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Parabola investigation

Edit 4: I added the below picture for clarity I'm trying to figure out how to find the angle between the red line and the blue line, but I have no idea how to start. (I have a feeling that this ...
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1answer
591 views

Find equations of the ellipses given conditions on the directrices, foci, and vertices

The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation: with distance between directrices $27$, and between foci $3$; with a focus at $(-\sqrt{13},0)$ ...
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Find the surface area of the part of the paraboloid $ z=5-(x^2 + y^2)$ that lies between the planes $z=0$ and $z=1$.

I have the following math question: Find the surface area of the part of the paraboloid $ z=5-(x^2 + y^2)$ that lies between the planes $z=0$ and $z=1$. So far i have computed $\sqrt{fx^2+fy^2+1}$ ...
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3answers
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Using trig identities to change from parametric to Cartesian equation

$$x=\sin t\\ y= 3\cos (3t)$$ Find $y$ in terms of $x$. I have graphed the function and it appears to follow $y(x)=-4x^2 +2$ from $-1\le x\le 1$ and $-2\le y\le 2$ . Thanks
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Conic sections and common functions

Is there a intuitive proof/reason of why plots of some common functions like y=x^2 are shaped like cross sections of a seemingly unrelated 3D object like a cone? ...
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68 views

Ellipsoidal Decomposition: Finding ellipsoids whose sum contains a given ellipsoid

We have a known ellipsoid $E\left(q,Q\right)$ in a 2D space. $q$ represents the center of the ellipsoid and $Q^{-1}$ is the weight matrix. The general equation of the ellipsoid is given as: ...
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I need some ideas regarding working model in mathematics. The topics are conics and vectors.

Hi can someone suggest me some working mathematical models under the topics conics and vectors. It should be 12th standard level. (Higher Secondary). I am trying to help my juniors to do this project. ...
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Polar correlation and conics in $\Bbb RP^2$

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
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1answer
15 views

Parabola equation expressed after x

Sorry for the bad title, as English is not my main language. Let me explain better what I mean. I have this equation of parabola: $y = x^2 + 4x $ What I want to do is get the $x$ in one side and ...
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1answer
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Calculating rotated half ellipse axis aligned bounding box

I want to find the height and width of a rotated half ellipse axis aligned bounding box. For that, I have the minor and major axes of the ellipse and its rotation angle. I have no idea how to form the ...
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Find the parametric equation of the following parabola?

It doesn't give me $2$ equations this time just $1$ and I have no clue what to do; $y^2 = 4x$ ANSWER IN BOOK: $x = t^2, y = 2t$
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1answer
601 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 ...
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1answer
71 views

Conic section: What is the coordinate matrix of its bilinear form?

Given is the conic section $x^2 + xy + y^2 + 2x +3y - 3 = 0$. I need to find the coordinate matrix $M_\beta(s)$ of the bilinear form $s: \mathbb{R}^2 \times \mathbb{R}^2 -> \mathbb{R}$. I read ...
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391 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
218 views

proof that intersection of two conic sections will intersect at at least two points.

In the following equation $\rho(x,y)$ returns a constant value for a given coordinate. $\mathbf n$ is the normal vector to the surface of the form $[P,Q,-1]$ and $s$ is a direction vector. ...
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1answer
22 views

Parabola - equation from three points

Question: Find the equation of the parabola whose axis is parallel to the y-axis and which passes through the points (0,4) (1,9) and (-2,6) Well as the parabola has its axis parallel to the y-axis ...
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2answers
57 views

Relation of ellipse semi-axes with rotation angle and projection length

In the following setup, assume $w$ (length of the projection of the ellipse) and $\theta$ (the rotation angle) are known. I want to know what equation(s) do I have here that helps me to derive the ...
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1answer
20 views

finding out wheter point is inside ellipse

I'm working on a way to determine if given point is "inside" given ellipse, the problem is I've already forgotten all the related mathematics and don't have time to relearn it and find a way to do it. ...
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3answers
37 views

How to find the outermost points in an ellipse?

If an ellipse is given in the form: $$ A(x − h)^2+ B(x − h)(y − k) + C(y − k)^2 = 1 $$ (where A, B, C, h, and k are given) What would be the simplest way of finding the outermost points, by which I ...
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Finding Equation of Hyperbola with given parameters

The Question Goes As:- Q.) Find the equation of Hyperbola whose eccentricity is at the origin, transverse axis along $x$-axis, length of conjugate axis is $5$ and passing through the points $(1,-2)$. ...
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1answer
900 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
810 views

Relationship between ellipsoid radii and eigenvalues

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the ...
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29 views

ellipse and segment intersection

I have a rotated ellipse, not centered at the origin, defined by x,y,a,b and angle. Then I have a segment defined by two points x1,y1 and x2,y2 Is there a quick way to find the intersection points?
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Calculating a Point that lies on an Ellipse given an Angle

I need to find a point (A on this diagram) given the center point of the ellipse as well as an angle. I've been melting my brain all day (as well as searching through questions here) testing out ...
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4answers
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How to calculate ellipse sector area *from a focus*

How do you calculate the area of a sector of an ellipse when the angle of the sector is drawn from one of the focii? In other words, how to find the area swept out by the true anomaly? There are ...
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1answer
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what are some applications of modern algebraic geometry to conic sections?

The simplest non-trivial example of an algebraic curve is probably a conic section (ellipse, parabola and hyperbola). At the same time, we also know that development in advanced theory can provide new ...
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1answer
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Conics Confusion

I'm currently reading through a document about the ellipse. I've attached the provided image and working out. From here, it is easy enough to show that $|OP|\sin\gamma=|FP|\sin\alpha$ using say the ...
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3answers
40 views

The line is tangent to a parabola

The line $y = 4x-7$ is tangent to a parabola that has a $y$-intercept of $-3$ and the line $x=\frac{1}{2}$ as its axis of symmetry. Find the equation of the parabola. I really need help solving this ...
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1answer
29 views

What is the equation of hyperbola

Given that the equation of asymptotes to the hyperbola be: $y=\pm\frac{3x}{2}$ and $b=4$ How to find the equation of hyperbola? I know that asymtotes have the equation $y=\pm\frac{bx}{a}$, ...
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2answers
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Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
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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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231 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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1answer
39 views

Locus on a parabola

How could I find the locus of M as P moves of the parabola. P is.(2at, at^2) . M is the midpoint of the x and y intercepts of the normal through P. So far I was able to find the quation of the normal ...
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1answer
280 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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1answer
304 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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2answers
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Degenerate conics

I was studying about the discriminant of a conic and got to the case where it equals 0. The book I'm referring to says that such a case means that the equation represents a parabola, a pair of ...
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How to draw the reciprocal polar of one ellipse w.r.t. another one

I know that the reciprocal polar of one ellipse w.r.t. another one is a hyperbola obtained as the envelope of all polars from points lying in one of the ellipses w.r.t. the other. My question is: how ...
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Catenary and parabola minimum comparison

Do the catenary and a parabola that approximates the catenary, have the same minimum (maximum sag)? IF plotted, it looks to me they do, and that they only difer somewhere on the "slope". (sorry for ...
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2answers
21 views

Ellipse as projection of a disk - function describing ellipse diameter with disk rotation?

Say I have got a disk of radius $r$ and a plane $p$ in $3D$ space. The disk is "aligned" to $p$ and lies at an arbitrary distance, so that its orthogonal projection on $p$ is an identical disk of ...
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2answers
683 views

Normal to Ellipse and Angle at Major Axis

I've tried to detail my question using the image shown in this post. . Consider an ellipse with 5 parameters $(x_C, y_C, a, b, \psi)$ where $(x_C, y_C)$ is the center of the ellipse, $a$ and $b$ are ...
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1answer
571 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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3answers
59 views

Solving $\frac{\mathrm d^2\mathbf{q}}{\mathrm dt^2} = -\frac{\mathbf{q}}{|\mathbf{q}|^3}$

I am reading a set of course notes and it has this example of a system of differential equations given by $$\frac{\mathrm d^2\bf{q}}{\mathrm dt^2} = -\frac{\bf{q}}{|\bf{q}|^3}$$ All it says is that ...
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How to draw a border at a specific distance from a cylinder outline

I have a small cylinder (Cylinder A) with its minor radius of A. Minor radius measures the minor radius of the cylinder ellipse. I need to draw a border at a distance of X from the border of ...
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Determining if two general conic sections are tangent to each other

Given two conics in general form $A_ix^2 + B_ixy + C_iy^2 + D_ix + E_iy + F_i = 0$ for $i = 1, 2$, I want to determine if they are tangent to one another, and I'm looking for a method that wouldn't be ...
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conic sections, ellipse [closed]

A particle is travelling clockwise on the elliptical orbit given by $$\displaystyle \frac{x^2}{100} + \frac{y^2}{25} = 1$$ The particle leaves the orbit at the point $(-8, 3)$ and travels in a ...
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1answer
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$P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ on the line that bisects $\angle F_1PF_2$. Prove $|PF_1-PF_2|>|QF_1-QF_2|$.

$\require{cancel}$ Sorry for the grammatical mistake in the title; it was needed to keep the title under 150 characters. $P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ is ...
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1answer
27 views

Finding equation of directrix when the parametric equation of parabola is given.

If the parametric equation of the parabola is $( x = t^2 + 1 , y = 2t + 1 )$, then find the equation of the directrix. This was the question in my last test in which I got stuck and wasted much of my ...
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39 views

finding $\lambda$ when equation of parabola is given

If the equation $\lambda x^2 + 4xy + y^2 + \lambda x + 3y + 2 = 0$ represents a parabola. Then find $\lambda$. I got stuck in this question while solving parabola. Is here anybody who can help me ...
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Equation of parabola with focus and tangent [closed]

What is the equation of parabola whose focus is $(1,1)$ and the tangent at vertex is $x+y=1$?
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Prove that the locus is a parabola

The point P(x,y) moves in XY plane such as that its distance from a fixed point (0,-1) is equal to its distance from the line Y=1. Prove that the locus is a parabola. Find it's focus, directrix, ...