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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1answer
71 views

Tangents of Rectangular hyperbola

P,Q,R are points on a rectangular hyperbola, and PQ perpendicular to PR. Prove that the tangent at P is perpendicular to QR.
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3answers
228 views

Generic rotation to remove Quadratic Cross-product

Show that if $b\neq 0$, then the cross-product term can be eliminated from the quadratic $ax^2 + 2bxy + cy^2$ by rotating the coordinate axes through an angle $\theta$ that satisfies the equation $$ ...
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1answer
334 views

Equivalence of geometric and algebraic definitions of conic sections

I have not been able to find a proof that the following definitions are equivalent anywhere, thought maybe someone could give me an idea: A parabola is defined geometrically as the intersection of a ...
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1answer
62 views

Polar equations of circles and ellipses

I have been trying to convert some conic sections from rectangular to polar form. I am fine going the other direction (given polar, convert to rectangular), but am having trouble going the opposite ...
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1answer
52 views

How so I put these in Standard form? Circle, Ellipse or Hyperbola?

I need help putting these into standard form so I can graph them. Also need help figuring out which ones are which: $$25x^2-16y^2-150x+64y-239=0$$ $$9x^2+4y^2+54x-64y+301=0$$ $$x^2+y^2-6x+8y+3=0$$
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6answers
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How to geometrically prove the focal property of ellipse?

How to prove geometrically that if we have a tangent of ellipse with focus F and F' in point P, that tangent is bisector of the angle between a line joining focus F to point P and the line F'P outside ...
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2answers
29 views

Explanation of graphical mathematical anomaly (for me, anyways)

I was working on some competition stuff when I came across the equation $y^2+2xy-x^2 = 0$, and the thing that surprised me was, when I graphed it, I got these two perpendicular lines at the origin, ...
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2answers
139 views

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
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0answers
103 views

What's the standard form of the equation of a line of a slanted parabola?

I have been trying to figure out the general form of a slanted parabola, but what I've gotten doesn't look like it would be accurate:$$(x-h)^2+(y-k)^2=\dfrac{d}{\sqrt{h}}$$Where $(h,k)$ is the focus, ...
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0answers
88 views

Real world application of slanted conics (parabolae especially)

I am writing a report on slanted conics of the form $$(x-h)^2+(y-k)^2= \dfrac{d}{\sqrt h}$$ Where $(h, k)$ is the focus, and $d$ is the directrix. Are there any real world applications for slanted ...
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1answer
128 views

angular velocity around ellipse

If I have velocity at perihelion/apphelion, distance away from sun at perihelion/apphelion, and orbital period. How can I find the angular velocity function for earth and subsequently all the other ...
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1answer
122 views

Classifying complex conics up to isomorphism as quotient rings of $\mathbb{C}[x,y]$

This is a continuation of the question I asked here. The problem is now: Let $Q = ax^2 + bxy + cy^2 + dx + ey + f \in \mathbb{C}[x,y]$ be a general quadratic polynomial, that is, $a,b,c \not= 0$. ...
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2answers
249 views

Find the Locus of the Orthocenter

Vertices of a variable triangle are $$(3,4)\\ (5\cos\theta,5\sin\theta) \\ (5\sin\theta,-5\cos\theta) $$ where $\theta \in \mathbb R$. Given that the orthocenter of this triangle traces a conic, ...
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3answers
267 views

Calculate tangent point on ellipse

I'm trying to find a tangent point on an ellipse. Trying a lot, using answers found a.o. on this site, but obviously doing something wrong as I'm not getting any good results. I've added a sketch, to ...
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2answers
384 views

Help creating equation for parabola word problem?

The cables of a suspension bridge create a parabola. The towers are 600 feet apart and 80 feet tall. If the cable touches the road halfway between the towers, what is the height of the cable at a ...
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1answer
83 views

Ellipse inscribed angles

On wikipedia in German, we find relations about two angles inscribed on parable and on hyperbole. The 4 points of the parabola $y = ax^2 + bx + c $ has the following property: $$ \frac{(y_4-y_1)}{(x_4-...
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1answer
465 views

Vertex Equation of an inverse quadratic function.

I'm working on a graphing web tool using JSXGraph, The user should be able to draw different functions. I was able to allow the user to draw quadratic functions by creating the vertex of the function ...
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0answers
35 views

Triangles with vertices on conics and their foci

Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that $$...
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2answers
114 views

Ceva, Desargues and Pascal's theorems for conics

I was told in class today that these three theorems are valid in projective geometry and with conic sections (I'm taking a modern geometry class) but I can't seem to find proofs anywhere online, and I'...
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2answers
68 views

Indefinite integral with sector of ellipse

An ellipse is given by the following equation: $$ 152 x^2 - 300 x y + 150 y^2 - 42 x + 40 y + 3 = 0 $$ After solving for the midpoint we have: $$ 152 (x-1/2)^2 - 300 (x-1/2) (y-11/30) + 150 (y-11/30)^...
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0answers
85 views

Maximum product of lengths involving secant drawn to a parabola.

A chord is drawn from a point $P(1,t)$ to the parabola $y^2=4x$, which cuts the parabola at $A$ and $B$. If $PA\cdot PB=3|t|$, what is the maximum possible value of $|t|$? All I can infer is that the ...
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1answer
84 views

Proving a statement about ellipses and Dandelin spheres.

I have the Dandelin sphere construction. That is, I am given a vertical cylinder with radius $r$ and two spheres of radius $r$ are put inside of it. A plane (horizontal or otherwise, just not vertical)...
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1answer
4k views

Find Tilted Parabola Equation given vertex and angle

How to find the parabola equation like the picture below, given the vertex $(x$$_o,y_o)$ and theta orientation? please help. thankyou.
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2answers
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I got this “parabolic” curve from a book but cannot find the right equation for it

The diagram below is taken from a book on Indian Stupa architecture. It says that the profile is a "parabolic" one. I have tried y=x^2 and varied the domains ox x and y but couldn't find the right ...
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2answers
107 views

Ellipse and rectangle

An ellipse, whose equation is ${x^2\over9} + {y^2\over4} = 1$, is inscribed within a rectangle whose sides are parallel with the coordinate axes. Another ellipse is circumscribing the rectangle and ...
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1answer
62 views

Interpreting 3D parametric equations

I've been working through a problem and I have managed to reduce it to the following:$$x=\frac{2r}{3}\cos\theta - \frac{r}{3}\sin\theta$$ $$y=\frac{2r}{3}\sin\theta - \frac{r}{3}\cos\theta$$ $$z = -\...
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1answer
85 views

The focal chord that cuts the parabola $ x^2 = -6y$ at $(6, -6)$ cuts the parabola again at $X$

The focal chord that cuts the parabola $x^2 = -6y$ at $(6, -6)$ cuts the parabola again at $X$. Find the coordinates of $X$. I have been going insane someone please help me :(
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2answers
223 views

Calculating semi axes from given tilted ellipse equation

Hopefully no duplicate of Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline? (see below) Let the following equation $$x^2 - \...
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0answers
65 views

Dandelin spheres and the asymptotes of a hyperbola

The other day, I was reading up on the synthetic geometry of conic sections a bit, and I wondered: is it possible to construct the hyperbola's asymptotes given just the intersecting plane and the ...
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0answers
39 views

Area swept out by non-solar focus not same over equal time?

Per Kepler's laws, the area swept out by a line between the sun and a planet is equal for a given period of time. The sun is also one focus of the planet's elliptical orbit. What about the area swept ...
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2answers
274 views

Fit an ellipse with constraints

I'd like to fit an ellipse with the equation of $ x^2 + ay^2 + bx + c =0 $ This is basically the equation of an ellipse with no tilt and with its center on the horizontal axis. I have some ...
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2answers
110 views

Ellipse with center in origin

The purpose is to fit data to a ellipse which center is the origin $(x_0=0,y_0=0)$. I found the general quadratic curve: $$ax^2+2bxy+cy^2+2dx+2fy+g=0$$ Reference: http://mathworld.wolfram.com/Ellipse....
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1answer
60 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
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4answers
715 views

Show that if an ellipse and a hyperbola have the same foci, then at each point of intersection their tangent lines are perpendicular.

I have to show that: If an ellipse and a hyperbola have the same foci, then at each point of intersection, their tangent lines are perpendicular. So I know that if I prove it for one of the ...
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1answer
64 views

Find a specific rectangle in an ellipse

For a software developpment, I need to find a rectangle that fits in an ellipse. I have an outer rectangle (left, top, width and height) and a function that draws an ellipse in it. Now I need to know ...
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1answer
196 views

Rolling ellipse on line - tangent and normal of roulette

Suppose that an ellipse is rolling along a line. If we follow the path of one of the foci of the ellipse as it rolls, then this path formes a curve - namely an undulary. Now consider the following ...
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1answer
41 views

Grade 10 Quadratic equation

This was on my year 10 maths test and I gave up with 40 mins to complete: Basically you were given the coordinates: y intercept : (0,10) 1 x intercept: (10,0) and y value of the vertex: +15 Can ...
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9answers
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Why is an ellipse, hyperbola, and circle not a function?

I am aware of the vertical line test. If you place a vertical line over a shape, and if it crosses more than once, it fails the vertical line test and is no longer a function. But I don't understand ...
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1answer
27 views

How to convert formulas for different standard parabolas?

There are 4 types of standard parabolas , and I'm supposed to remember many formulas about them like tangent , normal etc. But the problem is , if i know a certain formula for $y^2=4ax $ how can i ...
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1answer
33 views

help needed in understanding general conics proof

The origin is a centre of a general conic of second degree iff the coefficients of linear terms vanish. $ (\Rightarrow)$ part: Let $$ Q(x,y)\equiv ax^{2}+2h xy+ by^{2} + 2gx+2fy+c=0$$ books ...
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1answer
247 views

Arc length of parabola between two points

Well lets take a parabola of the equation $y = f(x)$ where $f(x)$ is obviously a $2^{nd}$ degree function. Now lets take two points at $x=a$ and $x=b$ . So can anyone please help me to find that ...
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1answer
93 views

Focus of a parabola, without derivatives

I have a seemingly easy question, but I have no clue how to find out its answer. I have the function $$f(x)=\tfrac{1}{8} x^2$$ This function is for (a parabolic cross-section through) a paraboloid ...
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1answer
101 views

transformation of conic section

Given is the conic section $x^2 +xy + y^2 +2x +3y -3 = 0$. The following tasks: 1.) What is the coordinate matrix $A_1 = M_{\beta} (\sigma) $ of the bilinearform? 2.) do the transformation and ...
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2answers
55 views

Better substitution calculating integral?

I'm calculating $$ \iint\limits_S \, \left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}} \right)^\frac{1}{2} \, dA$$ with $$S =\left\{ (x, \, y) \in \mathbb{R}^2 : \frac{...
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1answer
432 views

Given area of sector and a starting angle from focus of an ellipse, finding angle needed to get area.

Problem Background: I'm trying to make a rough simulation of Kepler's second law (equal areas over equal time) and to do this I've divided the area of the ellipse into some number of pieces. I want ...
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1answer
63 views

Conic involving circle question.

The question is: If the curves $ax^2+4xy+2y^2+x+y+5=0$ and $ax^2+6xy+5y^2+2x+3y+8=0$ intersect at four concyclic points then the value of a is???? The options are: a) 4 b) -4 c) 6 d) -6 I've ...
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1answer
50 views

Graphing Circles, Ellipses, Parabolas, and Hyperbolas

I need help plotting a curve on a graph where the distance from focus1 is always the same ratio to the distance from focus2. For instance, lets assume focus1 is -5 along the x axis, and focus2 is +5 ...
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5answers
259 views

Find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$

How to find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$ Let $y^2-xy-2x^2 =0...(1)$ and $y^2=x-2...(2)$ In equation (1) coefficient of $x^2 =-2; y^2=1, 2xy =\frac{-1}{2}$ ...
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3answers
216 views

Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline?

Given the ellipse $$3x^2-x+6xy-3y+5y^2=0$$ find the following: semi-major axis, $a$ semi-minor axis, $b$ displacement of centre from origin (or coordinates of centre of ellipse $(h,k)$) angle of ...
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3answers
129 views

Find equation of tangents to hyperbola

$$\frac{x^2}{4} - \frac {y^2}{16} = 1$$ There is a point $(1,2)$ where $2$ lines pass through and are a tangent to both curves. How do I find the equation of both lines?