Questions tagged [conic-sections]

For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.

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1answer
38 views

Given an ellipse and a reference point, how to find the two lines that are tangent to the ellipse?

I have an ellipse, possibly rotated and shifted from the origin, which is given by a parametrization similar to this one: $$ \begin{aligned} x &= x_0 + a\cos\theta\cos\alpha - b\sin\theta\sin\...
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0answers
111 views

The length arc of parabola

Find the length of the arc of the parabola $y^2=4x$ from $x=0$ to $x=4$. In the manual solution is $2\sqrt{5}+\ln(2+\sqrt{5}).$ My answer is $\displaystyle 2\int_0^4 \! \sqrt{1+\frac{\mathrm{d}x}{\...
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0answers
33 views

Why standard equation [closed]

We know standard equation of a parabola $y^2=4ax$ is found by considering focus to be $(a,0)$ and directrix to be $x+a=0$. But focus is just a fixed point and directrix is a fixed line so we can ...
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2answers
39 views

Breaking down the equation of conic section

This is a second degree conic section in 2 variables . It's an equation of a eclipse .In this question we are asked to find the centre , eccentricity and the length of axes. I was able to get the ...
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1answer
57 views

Why tangents to a quadratic curve never cut it again?

I have been studying conic sections lately and couldnt figure out why we can use the condition D=0 for the quadratic equation having only one root in many of the questions involving finding tangents, ...
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1answer
49 views

Common normal for an ellipse and parabola

I came across the following question in a magazine with example problems for the JEE examination in India: Given are two curves $x^2/a^2 + y^2/63=1$ and $y^2=4x$. The maximum integral value of $a$ ...
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1answer
36 views

Conic sections ( coordinates of the point of intersection of tangents with the curve )

I was told by my math teacher that equation of a pair of tangents drawn from the point is $T^2=SS_1$ This gives a second degree equation in $X$ and $Y$, which represents a pair of straight lines ...
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1answer
42 views

Classification of quadratic rational Bézier curves

My teacher months ago gave me a few hints on a method that can classify quadratic rational Bézier curves as different conic sections (arcs of those). As I recall, it starts with such a curve given ...
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1answer
44 views

Why is this solution incorrect?

Prove that on the axis of any parabola there is a certain point K which has the property that, if a chord PQ of the parabola be drawn through it, then $\frac{1}{PK^2} + \frac{1}{QK^2}$ is the same ...
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1answer
32 views

What are the main steps of determining the inequality by its given graph?

What are the main steps of determining the inequality by its given graph? How can I determine the following inequality? I am not asking for the ready answer, I just want to know which major steps in ...
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1answer
39 views

Name or Adjective for Ellipse with very different (or very similar) scales

I am looking for an adjective or word to describe an ellipse (or ellipsoid, in more dims) where the length of the principal axes are of roughly the same order of magnitude $\mathcal{O}(a) = \mathcal{O}...
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48 views

How to express by a single equation a region delimited by ellipsoid and hyperboloid

This is a follow-up to the earlier question about finding the equation for the region delimited by confocal ellipse and hyperbola. That was in $2$D and now I want to obtain an equation in $3$D. Let us ...
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1answer
85 views

Calculate the area of an triangle that is inscribed into an ellipse such that the elliptical sectors are of equal size.

Let us inscribe a triangle $ABC$ into a ellipse such that the sectors $S_1$, $S_2$ and $S_3$ have an equal sized area. This situation is depicted by the figure below. How we can calculate the triangle'...
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2answers
57 views

Locating the vertex of a hyperbola from a particular function

I've made a function for my physics research. My advisor wants all of the parameters to have a clear physical meaning. I've been calling one of the parameters "transition interval" but a ...
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2answers
156 views

What does partial differentiation give for a second degree equation which doesn't represent a conic?

Question says: Find real solution to the equation $$3x^{2}+3y^{2}-4xy+10x-10y+10=0.$$ My first thought was to treat it as a general conic ( $ax^2 + 2hxy + by^2 + 2gx + 2fy + c=0 $) but when I do so, ...
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32 views

Rewriting affine quadric equation by a coordinate change

I have the projective quadric given by the equation $$Q(X,Y,Z)=YZ-Z^2$$ in $\mathbb{P}(V)$, where $V$ is a $3$-dimensional vector space, and would like to find the equation of the affine conic ...
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1answer
115 views

What is wrong with my equation for an ellipse?

I'm drawing an ellipse on the screen by generating lots of random y-values for every x-value (this creates bristles of a paintbrush I can later use to draw strokes). Since wikipedia says the equation ...
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3answers
91 views

How to solve system of equations $A_1x^2 + B_1xy + C_1y^2 + D_1x + E_1y + F_1 = 0$ and $A_2x^2 + B_2xy + C_2y^2 + D_2x + E_2y + F_2 = 0$? [closed]

How to solve the system of equations? $$ \begin{cases}A_1x^2 + B_1xy + C_1y^2 + D_1x + E_1y + F_1 = 0 \\ A_2x^2 + B_2xy + C_2y^2 + D_2x + E_2y + F_2 = 0 \end{cases} $$ How to express x through y?
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2answers
177 views

A square ABCD has all it's vertices on $x^2y^2 = 1$. Midpoints of it's sides also lie on the same curve . Can I show diagonals of it meet at origin? [closed]

A square ABCD has all it's vertices on $x^2y^2 = 1$. The midpoints of it's sides also lie on the same curve . What is the area of this square ? It can be found very easily if we can show that diagonal ...
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1answer
58 views

Coordinate geometry and Trignometry.

Find the condition so that the line $px +qy=r$ intersects the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points whose eccentric angles differ by $\frac{\pi}4$. Though I know how to solve it using ...
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0answers
80 views

How to solve $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ if $x_{1,2} = x_0 \pm \sqrt{R^2 - (y - y_0)^2}$?

I'm trying to find the intersection points of circle and conic section curve. So, I'm solving the system of equations: $$ \begin{cases}Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \\ (x - x_0)^2 + (y - y_0)^2 =...
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0answers
20 views

Intersection of an ellipsoid given by SPD matrix with axes

Given a SPD matrix $\mathbf{A}$, for example $\mathbf{A} = \pmatrix{ 5 & 4 \\ 4 & 5}$ defining an ellipsoid $\epsilon_\mathbf{A}$ with semiaxes length given by $\frac{1}{\lambda_i} \mathbf{v}...
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1answer
48 views

Is $x^2=4ay$ a function while $y^2=4ax$ is not? [closed]

I just want to know if $x$ is always the independent variable.
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1answer
43 views

Cartesian equation of uniformly accelerated motion

Problem Let $\xi,\eta$ be the horizontal and vertical coordinates of a plane. Consider the following sequence of point \begin{equation}\begin{aligned} \xi_k &= \xi_0 + kT \dot{\xi}_0 + k^2 \frac{T^...
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0answers
35 views

Degenerate Conics Criterion

Suppose we have a quadratic polynomial equation, $f(x,y)=0$. How do we determine if the conic, over $\mathbb{R}$, is degenerate? Over $\mathbb{C}$ the answer is much simpler. We can replace $f$ by $F(...
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1answer
57 views

The minimal value of a fraction based on a focal chord of an ellipse [duplicate]

I came across a very interesting olympiad problem. It goes as follows: suppose you have an ellipse given by $\frac{x^2}{16}+\frac{y^2}{9}=1$ and a line that goes through the point $A(\sqrt{7},0)$. ...
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0answers
35 views

On a closure theorem involving 3 conics and an inscribed 3$n$-gon whose sides pass through 3 fixed points

(Note: I'm not a native English speaker.) When I was playing around with Geogebra, I personally discovered some interesting properties concerning conic sections. Here is one of them. Consider the ...
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1answer
39 views

Rotation matrix to construct canonical form of a conic

I want to find the canonical form of the following conic: $$C: 9x^2+4xy+6y^2-10=0.$$ I've found $C$ is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've ...
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0answers
68 views

Finding $3\tan\frac{p}2\tan\frac{q}2$, where $p$ and $q$ are the eccentric angles of the endpoints of a focal chord of $\frac{x^2}{16}-\frac{y^2}9=1$

If $PQ$ is a focal chord of hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$ , where eccentric angles of $P$ and $Q$ are $\alpha$ and $\beta$ respectively, then value of $3\tan \frac{\alpha }{...
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1answer
30 views

why do we always take the parametre of parametric equation of ellipse as the angle formed with x-axis instead of semi-major axis?

For ellipses having equation in the form of $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ why are the parametric equations are always $$x=a\cos(\theta)$$ $$y=b\sin(\theta)$$ even when b>a? As far as I know,...
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1answer
49 views

Is there a parametrization of a hyperbola $x^2-y^2=1$ by functions x(t) and y(t) both birational?

Consider the hyperbola $x^2-y^2=1$. I am aware of some parametrizations like: $(x(t),y(t))=(\frac{t^2+1}{2t},\frac{t^2-1}{2t})$; $(x(t),y(t))=(\frac{t^2+1}{t^2-1},\frac{2t}{t^2-1})$; $(x(t),y(t))=(\...
3
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2answers
322 views

Algorithm to determine if a 3D ellipsoid is contained within another?

Can anyone point me to an algorithm for how to efficiently check if a 3D ellipsoid is contained within another one? We can assume their origins are collocated. I am dealing with covariance ellipsoids ...
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1answer
43 views

Find domain and range of the slanted hyperbola

Given the conic section $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ and I know that it is a hyperbola and $B\ne 0$. How to find its domain and range? I guess the method of Lagrange multipliers will fail here.
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5answers
216 views

Eliminate $t$ from $h=\frac{3t^2-4t+1}{t^2+1}, k=\frac{4-2t}{t^2+1}$

Eliminate the parameter $t$ from $$h=\frac{3t^2-4t+1}{t^2+1}$$ $$k=\frac{4-2t}{t^2+1}$$ This is not the actual question. The question I encountered was: Perpendicular is drawn from a fixed point $(3,...
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1answer
29 views

Find angle of rotation of hyperbola given two asymptotes

Given two asymptotes $y=m_1x+b_1$ and $y=m_2x+b_2$, and a point on hyperbola $(p,q)$ is there a formula for finging an angle of rotation? I've found a formula (How do I find the slope of an angle ...
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1answer
19 views

Fitting a ballistic trajectory to noisy data where both spacial and temporal domains observations are noisy

Fitting a curve to noisy data is somewhat trivial. However it generally assumes that data abscissa is fixed, and the error is computed on the ordinate. In my setup, I have 3D spacial observations of ...
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2answers
41 views

Equation of a section plane in hyperbolic paraboloid

Find the equation of a plane passing through $Ox$ and intersecting a hyperbolic paraboloid $\frac{x^2}{p}-\frac{y^2}{q}=2z$ $(p>0, q>0)$ along a hyperbola with equal semi-axes. My attempt: The ...
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1answer
58 views

Conic chords projected by point $P$ form a quadrilateral whose vertex pairs are collinear with $P$

Starting with a conic $c$, a point $P$ not on $c$, and points $D,E,F,G$ on $c$, let $D',E',F',G'$ be the second points of intersection of the lines $PD,PE,PF,PG$ with $c$. Let $$ \begin{align} L &=...
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1answer
40 views

What can you say about the motion of an object with velocity vector perpendicular to position vector? Can you say anything about it at all?

I know that velocity is always perpendicular to the position vector for circular motion and at the endpoints of elliptical motion. Is there a general statement that can be made about the object's ...
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0answers
63 views

What is the shape of this curve?

Assume we have a cone, as shown on the left of the diagram, with its vertex at A and base a circle BC(C’)B passing through points B and C. If we cut it around the line AC, expand it out , it’s a ...
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2answers
86 views

Determining center of inscribed ellipse of a pentagon

Given a convex pentagon $ABCDE$, there is a unique ellipse with center $F$ that can be inscribed in it as shown in the image below. I've written a small program to find this ellipse, and had to ...
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2answers
80 views

What is the midpoint of the chord of contact form a point $P(4,5)$ to the curve $3x^2+4y^2=1?$

My method is to find the equation of the chord wrt to the point $P$ and the midpoint, say $O(\alpha,\beta)$ and then compare them. Equation of the the chord wrt to $P$ is$^1$: $$T=0 \text{ where } T=...
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1answer
40 views

Given an equation, determine what conic section the equation represents

I have the general equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ and I want to determine what conic section it represents according to the value of its coefficients or relations between them, is there a ...
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1answer
46 views

How to construct the foci of an ellipse given both its axes' support lines and two points on the conic

This is an easy problem to solve with analytic geometry, but not so easy with straightedge and compass alone. We're given two perpendicular lines which meet at the center $O$ of the ellipse $\mathcal ...
4
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3answers
116 views

Construct focii of ellipse given center and four tangent lines

We are given 4 distinct lines $a,b,c$ and $d$ which are said to be tangents to an ellipse. Let's consider that the 4 meetings of these lines form an convex quadrilateral $ABCD.$ There is a theorem ...
1
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1answer
29 views

Angle between normal vector of ellipse and the major-axis.

I am trying to derive the angle made between the major or x-axis and the normal vector of an ellipse of general shape $x = a\cos(t),y=b\sin(t)$ with the parameter $t$ reffering to Ellipse in polar ...
2
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3answers
67 views

Find a point $P_2$ on an ellipse, whose chord with $P_1$ is a max distance $d$ from its nearest side

I'm not sure if this solution is available in closed form, but after drawing it out I do think there will be two unique solutions always. I unfortunately have no clue where to start. Given: An ...
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0answers
48 views

find the conic curve through given points

I am trying to find a conic curve that passes through $[0:1:2]$ and intersects the conic curve $F =x_1^2 + 4x_1x_2 + 4x_0x_2 -x_0^2$ at the point $[1:1:0]$ with multiplicity $4$. Since I am given $5$ ...
2
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2answers
76 views

Is an equation only a parabola if it is quadratic? Could the graph of $y = x^{1.65}$ be described as parabolic in shape?

If given the equation $y = x^{1.65}$, could it be described as parabolic in shape, or does the equation have to have $x^2$ as its highest degree term?
2
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0answers
64 views

What is the difference between hyperbola and ellipse, since they both have the equation, $\frac{y^2}{a^2}+\frac{x^2}{a^2-c^2}=1$?

Noted that the equation of ellipse is given by $\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1$,where $b^2=a^2-c^2$ While the equation of hyperbola is $\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1$,where $b^2=c^2-a^2$ ...

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