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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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Showing that the area of a parabolic sector is half the area of a corresponding region bounded by the directrix (without Calculus)

Given parabola: It is necessary to prove that the area of the parabolic sector (green) is equal to half the area of the parabolic rectangle (orange) without calculus. ("Something like" ...
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1answer
48 views

Ellipse equations paradox

We know that an ellipse can be plotted in cartesian coordinates using the following parametric function: $$ ellipsePoint(\theta)=\left[\begin{array}{c}a \cdot \cos(\theta)\\b \cdot \sin(\theta)\end{...
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Fixed and variable Circle question

Two perpendicular normals to variable Circle are tangent to fixed circle $\ C_1$ with radius 2 and locus of centre of variable circle be the curve $\ C_2$, then find the product of maximum and minimum ...
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Tangents to different branch of hyperbola Mathematical proof

We can draw two tangents from an external point to a hyperbola The two tangents can be made on; The first branch only The second branch only One on the first and the other on the second branch. We ...
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46 views

Showing that an ellipse, as the intersection of cone and plane, matches the two-focus definition [on hold]

Conic sections are formed by the intersection of a cone an plane. But how do we know that the shape developed by the plane is the same which we define in other ways.what i mean is that suppose the ...
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24 views

Rotation between two ellipsoids?

Suppose I have two ellipsoid. One ellipsoid is inside of another ellipsoid. I know all 3 principal axis of both ellipsoid. Now, I can think of 3 possibilities that smaller ellipsoid can be placed ...
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1answer
21 views

Calculate the radius of a circle or sphere given a section?

This is probably a basic Math101 problem for most of you, but I'm not a mathematician so I could use some help with it. In the diagram below, how would I solve for R (the radius of the circle or ...
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1answer
35 views

Finding tangent angle

Finding the tangent angle between the negative $x$-axis and the parabola $$y=-ax^2+bx$$($a,b>0$) at $(x_0,y_0)$ : I am trying to find the tangent angle with negative $x $ axis for a parabolic curve....
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2answers
38 views

How can I create a seam for an ellipse?

I want to create a seam for an ellipse. (Mathematically I think this means that there are constant-length normal lines between the ellipse and the curve that creates the seam, but I'm not 100% sure ...
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1answer
48 views

Expression for the tangent points of a parabola with a point. Why does this work?

A couple of years ago I stumbled upon a curious expression that determines the tangent points to a parabola. Given a point $A(x_a,y_a)$ in $\mathbb{R}^{2}$ and a generic parabola $(p)$ $y = ax^{2} +...
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1answer
27 views

Prove the lines from the foci to a point on an ellipse form equal angles with any tangent vector at that point

Suppose we have an ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$ with parametrization $\gamma(t)=(p\cos(t),q\sin(t))$. Let $\vec{p}= \gamma(t_0)=(p\cos(t_0),q\sin(t_0))$ be any point on the ellipse. Let ...
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1answer
32 views

Find the equation of ellipse: Focus at $(-\sqrt{13}, 0)$, vertex at $(0,2)$ [closed]

The equation has its centers on the origin and their major axes on OX. I want to find the equation of the given Focus at $(-\sqrt{13}, 0)$ and vertex at $(0,2)$ .
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1answer
31 views

Prove That Area of Isoceles Triangle in an Ellipse is Maximum When Vertex On The Major Axis Lies On The Line Of Symmetry of the Triangle?

This is one of 101 classes questions whose solutions can be easily found on google, but most of the solutions assume without giving any proper line of reasoning that to maximize area (unique)vertex on ...
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47 views

generic parabola in polar coordinates

Starting from the equation $y=ax^2+bx+c$ substituting I get the next equation in polar coordinates: $$a\cos^2 \theta\ \rho^ 2 + (b \cos \theta - \sin \theta)\ \rho + c = 0$$ in case C was $0$ we could ...
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45 views

A weird function graph

https://www.desmos.com/calculator/wpfmd3l3ev If you see the graph in the above link, it will appear as if it under a parabola. Could someone explain what part of the function makes it appear like ...
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2answers
28 views

Find the locus of midpoints dropped from hyperbola to asymptotes

If $PN$ is the perpendicular from a point on a rectangular hyperbola $x^2-y^2=a^2$ on any of its asymptotes, then find the locus of the midpoints of $PN.$ On drawing a rough sketch of the graph, I ...
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2answers
69 views

Is the right intersection of an oblique circular cone an ellipse?

The bottom of a glass cup (assuming it is a true circle) looks like & is drawn as an ellipse. But is the shape we are seeing really an ellipse? After some tinkering with pen and paper, I see that ...
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1answer
32 views

Find equation of hyperbola, symmetric about the origin, traced by a point whose distance to point $(4,0)$ is twice its distance to line $x=1$.

The question is as follows: Find the equation of a hyperbola which is symmetrical about the origin traced by a point that moves so its distance from the point $(4,0)$ is twice as far as the point ...
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2answers
51 views

Why does Desmos plot $\frac1{50}(-\ln(\cot(\frac{x}{2}))(\cos x-1)+\frac12x^2-2\ln(\cos(\frac{x}{2})))$ as a dotted parabola?

$$\frac{-\ln\left(\cot\left(\dfrac{x}{2}\right)\right)\left(\cos\left(x\right)-1\right)+\dfrac{x^2}{2}-2\ln\left(\cos\left(\dfrac{x}{2}\right)\right)}{50}$$ https://www.desmos.com/calculator/...
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1answer
34 views

Find the locus of points $P$ for which angle bisector of tangents to a standard ellipse is $y = x \tan \theta$.

Find the locus of points $P$ for which angle bisector of tangents to a standard ellipse is $y = x \tan \theta$. Let $ P = (u, v)$ and ellipse be $E = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - 1$ Then ...
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2answers
45 views

Geometrical proof for length of chord passing through vertex of parabola

In a parabola $y^2=4ax$ , the length of focal chord making an angle $\theta$ with the x - axis is $4acosec^2\theta$ . If a chord is drawn parallel to that focal chord which passes through vertex of ...
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2answers
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How does a parabola sit smoothly between cartesian ellipse and hyperbola?

A parabola is supposed to sit between ellipse and hyperbola. And indeed, in the polar form $r=\frac\ell{1+e\cos\theta}$, we pass smoothly through a parabola when the eccentricity $e$ passes through $...
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1answer
49 views

Are the directrix line and focus unique for a conic?

How to prove of disprove the uniqueness of pair of (or single for parabola) directrix line and pair of focal points for a conics? (Ignore the degenerate cases and the circle.)
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3answers
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Diameter of an Ellipse at an Angle

A standard ellipse with semi-major axis $a$, semi-minor $b$ has a "diameter" of $2a$ in one dimension ($\phi=0$) and $2b$ in the other ($\phi=\pi/2$). Is there a function to find the diameter for an ...
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2answers
72 views

Line through $B=(2,5)$ meets $2x^2 − 5xy + 2y^2 = 0$ at $P$, $Q$. Find locus of $R$ on line such that $BP$, $BR$, $BQ$ are in harmonic progression

A variable line $L$ passing through the point $B(2, 5)$ intersects the crossed lines $$2x^2 − 5xy + 2y^2 = 0$$ at $P$ and $Q$. Find the locus of the point $R$ on $L$ such that distances $BP$, $BR$, ...
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2answers
29 views

Find the unit normal to an ellipse given by an equation

The equation of the ellipse is given as being: $$x^2 -xy + y^2 = 7$$We're instructed to find a unit normal to the curve at a general point $P(x,y)$, and also at point $(-1,2)$ in particular. My ...
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1answer
44 views

Conic sections: Parabola - What is $p$?

Help my teacher says $p$ can't be negative because it's distance. I watched TOCT's tutorial (The Organic Chemistry Tutor) in YouTube about parabola and he said "$(x-h)^2 = 4p(y-k)$ if $p$ is ...
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1answer
26 views

Optimizing the surface area of an ellipsoid [closed]

How can I optimize the surface area of an ellipsoid to make it as large as possible, while keeping volume fixed? After I rearranged the volume of an ellipsoid to solve for one of the radii, I tried ...
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2answers
51 views

Locus of the vertex of a variable parabola

A fixed parabola $y^2 = 4 ax$ touches a variable parabola. Find the equation to the locus of the vertex of the variable parabola. Assume that the two parabolas are equal and the axis of the variable ...
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1answer
51 views

Why are the axis of an ellipsoid eigenvectors?

Consider an ellipsoid $\{x| x^TAx = 1\}$. Let $A$ be a real symmetric matrix, and consider the eigen-decomposition $A=P\Lambda P^{-1} =P\Lambda P^T$, where the matrix $P$ is orthogonal because $A$ ...
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0answers
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Why is the eccentricity of a catenary equal to sqrt(2)?

I understand that eccentricity defines how circular a conic section is... but a catenary isn't a conic section - so why is its eccentricity equal to that of a rectangular hyperbola: $\sqrt 2$ ? How ...
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3answers
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How to find the equation of the conic before applying the rotation?

Given the rotation matrix: $$Q=\begin{bmatrix}\frac{2}{\sqrt{5}}& \frac{1}{\sqrt{5}}\\-\frac{1}{\sqrt{5}}& \frac{2}{\sqrt{5}}\end{bmatrix},$$ I want to find the equation of the conic $C$ given ...
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When are cone geodesics planar

I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section. One asked whether if you 'unroll the cone' the conic section becomes a straight line ...
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Ratio of axes in an approximate circle

I have some shape that is approximately circular and has area = 100 pixels, with every pixel having area 1. Is there some mathematical way I can define the ratio of the longest axis to the smallest ...
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2answers
62 views

A cone with guiding curve $x^2+y^2+2ax+2by=0$ contains $(0,0,c)$. Its section by $y=0$ is a rectangular hyperbola. Prove its vertex lies on a circle.

A cone has its guiding curve to the circle $x^2+y^2+2ax+2by=0$ and passes through a fixed point $(0,0,c)$. If the section of the cone by plane $y=0$ is a rectangular hyperbola. Prove that the vertex ...
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2answers
48 views

$Ax^2+Bxy+Cy^2=1$ with $A=C=1$ and $B=2$ should be a parabola (because $B^2 = 4 AC$). Instead, it represents parallel lines. What went wrong?

My calculus book says The equation $Ax^2+Bxy+Cy^2=1$ produces a hyperbola if $B^2>4AC$ and an ellipse if $B^2<4AC$. A parabola has $B^2=4AC$. When I set $B=2,A=C=1$, the equation becomes $...
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4answers
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Given a chord, how do I find the ellipse?

It will explain my use case at the end, in case I am approaching this wrong, but I will start with the math question. Given: a point $\rm P$ on an ellipse; the slope of the tangent (or normal) to ...
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61 views

Proof that the foci of an ellipse are unique

Given a fixed distance $2a$, and two points $(F_1,F_2)$ in the Euclidean plane, one can define an ellipse as the set of points $E$ such that the sum of the distances $d(E,F_1) + d(E,F_2)$ is equal to $...
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1answer
38 views

What is the equation for an ellipse in standard form after an arbitrary matrix transformation?

Suppose I have a general ellipse parameterized by a center point $(h, k)$, semimajor axis $a$, semiminor axis $b$, and rotation angle $\theta$. This has the formula in standard form of $\frac{((x−h)\...
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1answer
39 views

Rewrite general form of ellipse to standard form (what happened in step 3?)

From my math book (Rewriting general form to standard from) General form: $$8y+4y^{2}-18x+9x^{2}=23$$ $$-18x+9x^2 +8y+4y^2 =23$$ What happened from step 2 to 3? $$9(x-1)^2 -9+4(y+1)^2 -4=23$$ $$ ...
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3answers
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How to determine the area of a rotated ellipse?

The ellipse $6x^2+4xy+5y^2+8x+8y+1=0$ is neither expressed in terms of $x$; like $y=\pm\sqrt{a^2-x^2}$, nor in terms of $y$; like $x=\pm\sqrt{a^2-y^2}$. Separation of $x$ (or $y%$) may be impossible. ...
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6answers
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Find shortest distance from the parabola $y=x^2-9$ to the origin.

Find shortest distance from the parabola $y=x^2-9$ to the origin. First, I find minima of $\sqrt{x^2+(x^2-9)^2}$, so use derivative and ... Is have an easier way?
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2answers
71 views

Equation of auxiliary circle of the ellipse $2x^2 +6xy + 5y^2$ =1

Equation of auxiliary circle of the ellipse $2x^2 +6xy + 5y^2$= 1 My approach is , First I try remove xy term from the equation, to convert the given equation in the standard equation of ellipse and ...
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Can you construct a hyperbola with only the eccentricity, two axes of symmetry and semimajor axis length given?

I am trying to construct a hyperbola for a project I'm doing and I have the two axes of symmetry, the length of the semimajor axis and the eccentricity. Is it possible? If so, how?
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1answer
55 views

How to show $|F_1-v|+|F_2-v|=c$ for an ellipse with foci $F_1, F_2$

I am asked to show the sum of the distances between the two foci on the ellipse and any point, $v$, on the ellipse is independent of one’s choice of $v$. Starting with the standard equation of an ...
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1answer
24 views

Find $m$ such that the line is normal to the given hyperbola

Find $m$ such that $y=mx+\frac{25}{\sqrt3}$ is normal to $$\frac{x^2} {16}-\frac{y^2}9=1$$ How to go about this question. I don't find any clue.
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1answer
48 views

Find the directrix of the parabola with equation $y=-0.5x^2+2x+2$

Find the directrix of the parabola with equation $$y=-0.5x^2+2x+2$$ I did this: $$a=-0.5, b = 2, c = 2$$ Formula for the directrix is: $$y=-1/(4a)$$ $$y=-1/(4\cdot(-0.5))=3.5$$ This is not ...
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1answer
54 views

(conic section) Find focus point given the equation of the parabola $ (3x^2-6x+2) $

Judging by the picture, I don't think the focus point is correct, as it outside of the parabola. I derived the coordinates of the parabola by first determining the a, b, and c value. $$a=3, b=(-6), ...
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0answers
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Find a projective change of coordinates

Find a projective change of coordinates that takes the projective completion of the circumference C: $x^2 + y^2 = 1$ to the projective completion of the parabola P: $y^2=2px$, $p \geq 0$ (i.e. $...
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2answers
92 views

Find the locus of the foot of perpendicular from the centre of the ellipse.

Find the locus of the foot of perpendicular from the centre of the ellipse $${x^2\over a^2} +{y^2\over b^2} = 1$$ on the chord joining the points whose eccentric angles differ by $π/2.$ My approach ...