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Questions tagged [conic-sections]

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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Given orthogonal vibrations, how can I find the magnitude and direction of the major axis of the resulting ellipse?

The Context In my work, I am using a vibration table to test the resonant properties and survivability of a structure under different shaking regimes. The table is driven by an eccentrically weighted ...
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2answers
41 views

How to find the auxiliary circle of a non-standard ellipse?

Given the equation of conic C is $5x^2 + 6xy + 5y^2 = 8$, find the equation S of its auxiliary circle? Now, I know that the equation C is an ellipse. Since $\Delta = 5*5(-8) - (-8)(3^2) \neq 0$ And,...
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1answer
24 views

Verification on “concurrent points”, and a cubic discriminant

It is well known that three lines are said to be concurrent precisely when they all meet at a point, namely the point of concurrency. In the paper On Sets Defining Few Ordinary Lines (v3) by Green &...
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0answers
31 views

Other intersection of common tangents

Core question Suppose I have two conics, $A$ and $B$. In general these have four common tangents. Let $p$ be the point of intersection of two of these tangents, and $q$ the intersection of the other ...
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1answer
43 views

Is $y=x^2$ smooth at origin?

if $r(t)=t^2 i + t^4 j$, then it is a parabola $y = x^2$. It satisfies the condition of non smooth curve i.e. $\frac{dr}{dt}=0$ at $t=0$. But geometrically it shows the curve (parabola) is smooth at $(...
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0answers
39 views

Geometry problem on parabola and circle [closed]

The parabola $y=x^2$ touches a circle of radius $\sqrt{1+4t^2}$ at the point $P(t,t^2)$. Prove that the locus of the center of the circle is $y=x^2+1$ or $y={{x^2}\over9}-1$.
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21 views

Parabola and Conics [closed]

The parabola has equation y^2 = 4ax , where a is a positive constant. The point P(at^2, 2at) lies on C. The point S is the focus of the parabola C. The point B lies on the positive x-axis and OB = ...
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3answers
47 views

Prove that the roots of the equation $2kx^2+5x-k=0$ are real and distinct for all real $k$. [closed]

Can you please help me find the proofs for the values of $k$ being real and distinct? Every time I tried to solve this equation, the answer I got had no real roots.
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4answers
56 views

Find the equation of the common tangents

Find the equation of the common tangents to the parabola $y^2=4ax$ and $x^2=4by$. My Attempt : $y^2=4ax$ is the equation of parabola with focus at $(a,0)$ $x^2=4by$ is the equation of parabola ...
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2answers
130 views

Rectangular Hyperbola - A Property of Normals

Consider the rectangular hyperbola $xy = c^2$. Normals at points $P,Q,R$ and $S$ on the curve are concurrent, and meet at point $O(h,k)$. Find $OP^2 + OQ^2 + OR^2 + OS^2$. I managed to solve the ...
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1answer
43 views

Finding co-ordinates and t-value for closest point between two parametric curves.

Find for the two parametric equations: $x_t1=10402 cos⁡(t/980) $ $ y_t2=11066 sin⁡(t/980)-t^2/(4.55×10^6 )$ $x_t2=11258 cos⁡(t/1120)$ $y_t2=10398 sin⁡(t/1120)$ Where t is time passed in seconds ...
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1answer
34 views

Given a hyperbola, determine whether any of its points lie within $\{x\in [0,1]\mid 0\leq y\leq 1\}$

Given $$ y=\frac{c_1 +c_2x}{c_3+c_4x} $$ Is there any test using the values of $c$ to see if between $x\in[0,1]$ there is at least one point where $y\in[0,1]$ other than just evaluating the function ...
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1answer
31 views

Prove that the roots of the equation $2kx^2+5x-k=0$ are real and distinct for all real values of $k$. [closed]

As stated above, could someone help me solve the problem in the title? I'm having trouble when it comes to understanding everything about it, but what I specifically need to know is how to convert ...
0
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1answer
55 views

Does there exist a formula to find the coefficients of this parabola?

For equation $y=ax^2+bx+c$, assume I know the value of the coefficient $a$, I know the value of $y$ at the parabola's vertex (though I do not yet know the $x$ at that point), and I am given a point $(...
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0answers
62 views

Prove that the semi Latus rectum of an ellipse is the harmonic mean of the segments of focal chord.

I am a 12th student. I found this property in a reference book, without its proof. So i tried to prove this myself but got stuck. Here's my attempt at the problem: Basically the question is to prove $$...
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0answers
34 views

Hilbert manifold decomposition into infinite-dimensional ellipsoids

Let $X$ be a Hilbert manifold and $\mathcal{U}_{\alpha}$ an open subet of $X$, with a local coordinate chart $(\mathcal{U}_{\alpha},\phi_{\alpha})$ such that $X:=\bigcup_{\alpha\in A}\mathcal{U}_{\...
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5answers
97 views

Solution to Putnam 2007 A-1

2007 A-1: Find the values of $\alpha$ for which the curves $y=\alpha x^2 + \alpha x + \frac{1}{24}$ and $x=\alpha y^2 + \alpha y + \frac{1}{24}$ are tangent to each other. My solution: Notice that ...
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2answers
39 views

Locus in the case of an ellipse.

Given that $ S $ is the focus on the positive x-axis of the equation$ \frac{x^{2}}{25} + \frac{y^{2}}{9} =1$. Let $P=(5 \cos{t}, 3 \sin {t})$ on the ellipse, $SP$ is produced to $Q$ so that $PQ = 2PS$....
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2answers
48 views

Common point between ellipse and tangent passing through external point

Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and a point $(u, v)$ not on the ellipse, I want to find two points that lie on the ellipse and on the two tangents of the ellipse passing ...
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1answer
44 views

A tangent to an ellipse makes angles $\alpha$ with major axis and $\beta$ with a focal radius; show that the eccentricity is $\cos\beta/\cos\alpha$.

If the tangent at any point of the ellipse make an angle $\alpha$ with the major axis and an angle $\beta$ with the focal radius of the point of contact, then show that the eccentricity of the ellipse ...
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3answers
52 views

A double ordinate of the parabola

A double ordinate of the parabola $y^2=2ax $ is of length $4a $. Prove that the lines joining the vertex to its ends are at right angles. What does double ordinate actually mean? Seeing that the ...
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4answers
60 views

Hyperbola asymptotes from conic general equation [duplicate]

If I have the coefficients of the following equation: $$AX^2 + BXY + CY^2 + DX + EY + F = 0$$ And I know it's a hyperbola, how can I get the equations for the asymptotes with respect to the ...
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1answer
26 views

Why does ratio of $2$ linear expression gives you a rectangular hyperbola

We were studying Calculus and some methods on how to find domain and range, when my teacher suddenly said, "FYI, the ratio of $2$ linear expression gives you a rectangular hyperbola Can someone tell ...
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1answer
35 views

Find equation for a hyperbola without asymptote

I have a hyperbola graph with points (3,4) and (8,3). The graph can be modelled using y = a / ( x - b) I need to write the equation when x = 3 and find the values of a and b. I don't know how to ...
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0answers
34 views

Parabola Equation from its trajectory points in a Video

I have a video of an object describing a ballistic trajectory (i.e. A ball being thrown) The video is filmed with an angle and not orthogonally to the moving object, the camera is calibrated. I am ...
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1answer
45 views

Inner and outer borders of ellipse

I have created an ellipse of constant thickness. What are the shapes for the inner and outer boundaries? I know they are not also ellipses. I was only able to draw the shape in geometry software, and ...
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1answer
37 views

Conic Section Equation from Michael Spivak's Book

So i've been reading Michael Spivak's Calculus lately and now i feel im stuck in his conic section equation, page 81. enter image description here What i dont understand is, how can the first ...
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1answer
29 views

how to know the axes of an ellipse after rotation.

I came around a question: $P=\begin{bmatrix}3 & 1\\1 & 3\\ \end{bmatrix}$. Consider the set S of all vectors $\begin{pmatrix}x\\y\end{pmatrix}$ such that $a^2+b^2=1$ where $\begin{...
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2answers
63 views

What is the equation for the right-side branch of $Axy + Bx + C y = D$?

I would like to know how to derive the formula for the right-side branch of $Axy + Bx + Cy = D$, where the constant and coefficients are positive integers, and expressed as an equation and in terms of ...
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1answer
43 views

Check if the three trajectories are indeed one and the same ellipse

Consider three time dependent points $P_1(t)$, $P_2(t)$ and $P_3(t)$ in the plane($\mathbb{R}^2$), such that: $\frac{dP_1}{dt}(t)=P_2(t)-P_3(t)$ $\frac{dP_2}{dt}(t)=P_3(t)-P_1(t)$ $\frac{dP_3}{dt}(...
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2answers
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Study of parabolas of the form $(ax + by) ^ 2$ + $2gx + 2fy + c = 0$

I am studying parabolas of the form $(ax + by) ^ 2$ + $2gx + 2fy + c = 0$. In the text which I am referring to it is mentioned that we can do the following manipulations: $$(ax + by) ^ 2 = -2gx - ...
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2answers
52 views

How to parameterize an ellipse?

I need to parameterize the ellipse $\frac{x^2}{2}+y^2=2$, so this is how I proceed: I know that $a=2$ and $b=1$ (where $a$ and $b$ are the axis of the ellipse), so I parameterize as: \begin{cases} x=...
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1answer
60 views

Show that any plane whose normal lies on cone $(b+c)x^2+(c+a)y^2+(a+b)z^2=0$ cuts the surface $ax^2+by^2+cz^2=1$ is rectangular hyperbola

Show that any plane whose normal lies on cone $(b+c)x^2+(c+a)y^2+(a+b)z^2=0$ cuts the surface $ax^2+by^2+cz^2=1$ is rectangular hyperbola My attempt: let $\frac {x}{l} = \frac {y}{m} = \frac {z}{n}$ ...
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2answers
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Points of tangency from a point M to an ellipsoid-plane intersection

Having an spheroid $S$ of semi-major axis $a$ in the equatorial plane in direction $x-axis$ and $y-axis$, and of semi-minor axis $b$ in direction $z-axis$. $(S): (\frac{x}{a})^2+(\frac{y}{a})^2+(\...
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1answer
59 views

Functions and Graphs - How to find equation of a straight line with only 1 given point

can anyone help me with this question especially the first one...I don't know how to find the equation of the line as i'm only given one point.
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1answer
67 views

Is there a way to use the equation of an ellipse to obtain the angle of rotation of the horizontal ellipse as shown in the figure?

I need to obtain the angle of rotation($\theta$) of the ellipse that would cause it to touch the circle centered at $(x_2,y_2)$. The values $d, c, s.$ and $m$ are known and the coordinates shown in ...
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3answers
57 views

Tangent lines to ellipse from external point [closed]

I'm given an ellipse with the equation $$\frac{x^2}{5^2} +\frac{y^2}{4^2} -1 = 0 $$ and I have to find the tangents to this ellipse passing through a point $P(10, -8)$.
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0answers
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Internal ellipse formed by vertices of a pentagonal star

$(A,C,D,E)$ are four fixed points on a fixed outer (red) ellipse and $B$ is a variable point lying between $A$ and $C$. Alternate vertices are joined to form a pentagonal star. (drawn on Geogebra). A ...
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1answer
61 views

Lampshade geometry and moving the source of light

I've been thinking about lampshade geometry and the hyperbolic outline of light formed when the lamp is against the wall. This happens when the source of light is at the centre between the lampshade. ...
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1answer
51 views

Analytic geometry, parabola

I know this is very easy for some of you here , but sadly for me it's not. So kindly, please help me. A. How to get the equation of a parabula who opens to the left and whose latus rectum is equal ...
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3answers
84 views

Rotation of general conic equation - angle $\theta$ quadrant

I need to identify the conic represented by the equation $$9x^2 -6xy +y^2 - 40x -20y + 75=0$$ The book provides the solution as below: $\tan 2\theta = -3/4$ Then says take $\tan \theta = 3$. I ...
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0answers
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Coordinates of tangent points from a point outside a spheroid (ellispoid of revolution)

The blue conics in the figure is an ellipse (it would be in a case a circle), that represents the tangency points, from the line drawn from the point $P$ to the ellipsoid surface in all direction, ...
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0answers
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Power of a point parabola theorem [duplicate]

The theorem: Let AB be a focal chord of a parabola, let C be the circle with diameter AB, let V be the vertex of the parabola. The power of V with respect to C independent of the choice of the focal ...
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3answers
54 views

Find all complex numbers satisfying $z\cdot\bar{z}=41$, for which $|z-9|+|z-9i|$ has the minimum value

My first attempt was to express $z$ as $x+iy$ and minimize the expression $\sqrt{(x-9)^2+y^2}+\sqrt{x^2+(y-9)^2}$ where $x^2+y^2=41$. That said, it seems to me that using the geometric ...
5
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3answers
67 views

Shortest distance from ellipse to line

What is the shortest distance between the ellipse $$\frac{x^2}{4}+y^2=1$$ and the line $y=\frac{-\sqrt{3}}{2}x+8?$ I tried solving it by using a line that is tangent to the ellipse and parallel to ...
3
votes
1answer
69 views

Area of the largest rectangle with sides parallel to $x$- and $y$-axes that can be inscribed in the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$

First off, I simplified the equation of the ellipse to get it to $$9x^2 + 16y^2 = 144$$ And then did further simplification to get it in terms of: $$y= \bigg(9 - \frac{9 x^2}{16}\bigg)^{\frac{1}{2}...
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1answer
22 views

How to find an isometry that maps the cartesian equation of ellipse to the cartesian equation of the canonical form of ellipse

In the euclidean space $E 2$ consider the ellipse $C$ with foci $F_1 = (1, 7)$ and $F_2 = (7, 7)$, passing through the point $(4, 3)$. (i) Find a cartesian equation of $C$. (ii) Find a cartesian ...
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votes
0answers
23 views

Proof for diametral plane

I have seen a statement in text book which is given as a passing reference. It says that diametral plane of surface $f(x,y,z) = 0$ is given as $l \frac{df}{dx}+m\frac{df}{dy}+n\frac{df}{dz} = 0 $ in ...
0
votes
1answer
31 views

How can generator of hyperboloid of 1 sheet cut principal elliptic section in 2 points

I am a beginner in this area. So if i make wording related mistakes pls correct me. Generator means line contained entirely in a surface. There is a problem in my textbook which goes like this --> "...