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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Equation of locus

Point P$(x, y)$ moves in such a way that its distance from the point $(3, 5)$ is proportional to its distance from the point $(-2, 4)$. Find the locus of P if the origin is a point on the locus. ...
2
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0answers
48 views

Show that a complex equation represents a circle

I'm having troubling understanding the answer to a question. The question is: If $\ v=1+i$ and $\ z=x+iy$, for any real numbers x and y: Show that the equation $\left|z-v\right|= \left|vz\right|$ ...
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0answers
28 views

Rotation and translation of a parabola

I have a specific conic (which I've worked out is a parabola) $16x^2 - 24xy + 9y^2 - 60x - 80y + 20 = 0$ I have to use a rotation through an angle $\theta$ where $sin\theta=4/5$ and $cos\theta = ...
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4answers
69 views

Non-symmetrical parabola

Sorry if this is a really simple question, but I was looking for an equation to produce a non-symmetrical parabola. (The left side of the parabola would have a different 'slope' than the right side of ...
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1answer
28 views

How to find smallest tangent ellipse giving multiple lines ?

This ellipse must be tangent to at least 4 lines and it must intersect the other lines. I've tried to use ellipses that are parallel to the x- and y-axis. I've done this by transforming the equation ...
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1answer
52 views

Tangents to the parabola

Quite a straightforward piece of maths I can't seem to get my head around here: The tangent to the parabola at the point $(4a, 4a)$ is given by what equation? Bearing in mind the parabola is ...
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2answers
44 views

How to measure the angle of the tangent to an ellipse?

If we consider the ellipse in the picture here How do we determine the angle $\lambda$ of the vector v (tangent at point x = 2 ,y = 3) with the line joining the center (10,0)and the point (2,3)? ...
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1answer
46 views

Tangent of a conic section

Let $$ax^2 + hxy + by^2 + gx + fy + c = 0$$ be the equation of a conic section I want to find the equation of tangent and normal at some point on the curve say $$(x_0 , y_0)$$ I know there is some ...
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1answer
58 views

Rolling of one ellipse on another ellipse of same size when initially touching each other at the end of their major axis.

I have a question related to conic section which i could not understood. The question is $E_1$:$ \frac{x^2}{a^2} +\frac{y^2}{b^2}=1(a>b)$ is a given ellipse. Another ellipse $E_2$: is of same ...
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2answers
52 views

Reflection of $x=1$ about $x+y=1$

A ray of light travels along the line $x=1$ and gets reflected by a mirror on $x+y=1$. Find the equation of the reflected ray. $$$$ I am to solve this problem using only ...
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1answer
58 views

Why 5 point determine a conic?

How to prove that any five points, of which no 3 are colinear, there is a single conic that passes through all of them ? (I have to start out with the equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ but i don't ...
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2answers
23 views

A circular paraboloid can be a elliptic paraboloid?

I'm aware of this similiar question: what is the difference between an elliptical and circular paraboloid? (3D) But I need help in a different way. In my calculus exam, I was asked to name the ...
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0answers
10 views

Spherical to Cartesian coordinate ellipsoid overlap

I have two geographic coordinates; latitude and longitude , separated by few meters. I need to draw an ellipsoid of same major and minor axes ,centered around the geographic co-ordinates I used ...
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2answers
21 views

Question on Straight Lines invoving Family of Lines and Angle Bisectors.

Lines $L_1$ ($a_1x+b_1y+c_1$) and $L_2$ ($a_2x+b_2y+c_2$) intersect at a point $P$ and subtend an angle $\theta$ at $P$. Another line $L$ makes the same angle $\theta$ with $L_1$ at $P$. Find ...
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2answers
45 views

What shape do we get when we shear an ellipse? And more generally, do affine transformations always map conic sections to conic sections?

What shape do we get when we shear an ellipse? Is it another ellipse (or circle in special cases)? Or is it some other shape which isn’t a conic section? I was under the impression that applying any ...
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2answers
59 views

Angle between x-axis and major axis of ellipse

The MathWorld resource here gives a formula for the counter-clockwise angle $\phi$ between the x-axis and the major-axis of an ellipse $a x^2 +2bxy + cy^2 + 2dx + 2fy + g = 0$ as But what do I do ...
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1answer
40 views

Two questions on finding the equation of a parabola word problem- Klein's Calculus: An Intuitive and Physical Approach

I am solving the following word problem "A high voltage cable is supported by two towers 2800 feet apart and 348 feet high. The cable hangs in approximately the shape of a parabola, and the lowest ...
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1answer
24 views

How to find the locus from a general point

Given a general point on a line, say $\left(\frac35x \ ; {2 \over 5}(x+3)\right)$ or any other general point, how can you find the locus as this point moves?
3
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3answers
104 views

Parametric equation - of a hyperbola

I know that the parametric equation for points on a hyperbola($\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$) is: $$x = a\sec \theta$$ $$y = b\tan \theta$$ However, what does the parameter $\theta$ actually ...
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2answers
83 views

Circles tangent to a parabola

For the past two weeks I was struggling with solving the following problem. Description of variables: $(x_n,y_n)$ - center point of the circle $C_n$ $r_n$ - radius of the circle $C_n$ Given the ...
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0answers
22 views

Tangent unit vector on ellipse

Given an ellipse $ E = \{ x | x \in (x-c)^T \cdot A \cdot (x-c)\}$ where $c$ is the center of the ellipse, how do we find the unit vector tangent to the point $\xi \in E$? I am doing this because I ...
0
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1answer
15 views

which point is definitly on this parabola?

Vertex of a parabola is on y axis. and the point (4,7) is on this parabola. which one of these points definitly on the parabola : (-4,7),(2,7),(0,11),(-2,7) or (0,-5) Let $(0,b)$ be the vertex ...
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1answer
36 views

Find an equation of the quadratic function with zeros at $(0, 0)$ and $(6, 0)$ with $f(5) = -15$

The Question is: write the equation of the quadratic function with zeros at $(0,0)$ and $(6,0)$ with $f(5) = -15$. So, I know how to get the equation from the zeros, but I am confused with what I am ...
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0answers
21 views

Rational points on this ellipses (with constraint)

Ellipse : x2 + y2/2 = 1 Constraint on solution: x > 1/sqrt(2) Is there any rational solution ( both x and y are rational) for this ellipse with given constraint. Side note : x = 1/3 , y = 4/3 is a ...
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0answers
20 views

Find foci and eccentricity of ellipse given either 5 points or its general equation [duplicate]

I'm considering an arbitrary, non-degenerate ellipse here, i.e., without assuming that it's centred on the origin or either axis, nor oriented at any specific angle. I know either 5 points on the ...
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0answers
15 views

How to calculate two ellipse is similar with the other?

I'm beginning work about shape.But I don't know,how to calculate 2 ellipse(2D) are similar. I have point(x,y) of ellipse and semi-axis. My first idea calculate aspect ratio of semi-axis to compare ...
2
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1answer
48 views

How to determine arc length of a section of an ellipse

I need help resolving a geometric hydrology related question for a storm drain pipe. The pipe is 6.91 ft x 5.35 ft. The equipment at the site is giving me real-time depth of water data, and with the ...
0
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1answer
42 views

Find the circle circumscribing a triangle related to a parabola [duplicate]

Consider the following lines $x-y-1=0$ $x+y-5=0$ $y=4$ The line 1 is the axis of the parabola, the line 2 is the tangent at the vertex to the same parabola, and the line 3 is another tangent to ...
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1answer
29 views

A question on the parabola..

Consider the following lines $x-y-1=0$ $x+y-5=0$ $y=4$ The line 1 is the axis of the parabola, the line 2 is the tangent at the vertex to the same parabola, and the line 3 is another tangent to ...
0
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1answer
31 views

If $4a^2+9b^2-c^2+12ab=0$,the family of straight lines ax+by+c=0 is concurrent at which point?

If $4a^2+9b^2-c^2+12ab=0$,the family of straight lines $ax+by+c=0$ is concurrent at which point? How to solve such problems.Hints please!
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1answer
64 views

how to determine the width (x) of a cross section in an elliptical storm drain pipe given a known depth (y)

I am a hydrologist working on a stage-discharge ratings curve for an elliptical storm drain pipe. The pipe is 6.91 ft x 5.35 ft. I have equipment at the site that gives me real-time readings of water ...
0
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2answers
46 views

Why is the equation of an ellipse (x/a)^2 + (y/b)^2 = 1?

I've seen many proofs online, but I can't really wrap my mind around it. Being a generalization of the circle, I thought its equation would be as easy to understand as the circle's. Turns out I was ...
0
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1answer
50 views

The focus and length of the latus rectum of the parabola $y = x \tan \theta - \frac{gx^2}{2u^2 \cos^2 \theta}$

$$y = x \tan \theta - \frac{gx^2}{2u^2 \cos^2 \theta}$$ Find the focus and length of the latus rectum of this parabola. Here $u$ is a constant. We also know tangent at the origin makes an ...
0
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1answer
20 views

If for any external point, exactly two tangents can be drawn to an algebraic curve, must the curve be a conic?

Yesterday, my teacher, while proving Poncelet's theorem, seemed to use the fact that if from any external point (external meaning, I assume $f(x,y)>0$ where $f$ is the polynomial of two variables ...
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2answers
27 views

Are all conic equations functions?

Just wondering why we have equations defined in the Cartesian coordinate for circles and ellipses: wouldn't graphing those shapes contradict the fundamental property of a function (i.e.: to each ...
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2answers
71 views

Plucker's $\mu$

In Jürgen Richter-Gebert's book "Perspectives on Projective Geometry", he talks about Plucker’s $\mu$ in Section 6.3. He says that this trick was used by Plucker quite often. Plucker's trick involves ...
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1answer
14 views

Simplification canonical Ellipse equation Polar to Cartesian form

Request your help to trap error, trying to covert an ellipse equation form polar to rectangular form, canonical. $$ \dfrac{p}{r} =1 - \epsilon \cos \theta \tag{1}$$ $$ p = r - \epsilon x \tag{2} ...
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2answers
40 views

On the intersection of the hyperbolic paraboloid with a bundle of planes.

Suppose we are in Euclidian space in 3 dimensions. I intersect a bundle of planes $\alpha(x-y) + z = 0$ with a hyperbolic paraboloid $x^2 - y^2 = 2z$ \begin{cases} \alpha(x-y) + z = 0 \\ x^2 - y^2 = ...
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1answer
65 views

minimum number of points needed to define a unique 2 degree curve

as the title says to find minimum number of points needed to define a unique 2 degree. i did it by thinking that in general equation of 2 degree $Ax^2 + By^2 + 2Gx + 2Fy + 2Hxy + C $ there are 6 ...
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2answers
40 views

On the cylinder that projects a curve on a plane.

Suppose I have in space the curve $C$ given by $(x,y,z) = (t,t^2,t^3)$ and I want to project this curve $C$ on the plane $\alpha$ that has equation $x+y+z=0$. Then I take every vector orthogonal to ...
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2answers
75 views

Problem Related To Auxiliary Circle Of Ellipse

Tangent at any point $P$ on an ellipse whose foci are $F_1,F_2$ meets the auxiliary circle of the ellipse at $B_1$, $B_2$. If $F_{1}P+F_{2}P=10$ and $(F_{1}B_{1}) \cdot(F_{2}B_{2})=16$, then ...
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1answer
41 views

still confused about conics

Hello I have tried to post a few questions to try to understand better some things in conics. I am still really confused and am looking for help to understand. I am especially confused about graphing ...
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0answers
16 views

Interconverting formulas for hyperbola and its conjugate

There are two different sets of formulas of things like focus,equation of directrix,LR,eccentricity etc. for a hyperbola and its conjugate hyperbola. It is confusing to remember the equations ...
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2answers
29 views

show that the orbit represented by the function r() is an ellipse

let $r(θ)=a(1-β^2)/(1+β\cos\theta)$ representing the distance from the Sun to a planet. With $0<β<1$, show that the orbit represented by this function $r(θ)$ is an ellipse described by ...
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3answers
47 views

How to find directrix of conic

Hello I am having some confusion with conic sections. For example, I am asked to find the directrix and eccentricity of the ellipse given by the formula $$\frac{x^2}{9}+\frac{y^2}{16}=1$$ So, what ...
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2answers
31 views

Having some problems with understanding conics and graphing (eccentricity)

Hello I am having some trouble understanding how to graph some conic equations, and especially what the value of eccentricity is given. for example, one of the questions given was Draw an ellipse ...
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1answer
51 views

How can we show polar coordinates r(theta) is an ellipse?

$r(θ) = a(1 − β^2)/(1 + β \cos θ)$ and I want to show this $r(θ)$ is an ellipse described by $\dfrac{(x+\sqrt{a^2 − b^2})^2}{a^2}+\dfrac{y^2}{b^2}= 1$, when $0<β<1$. How can we show this?
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0answers
81 views

Equation for Hyperboloid of two sheets

Enter values $(A,B,C$)so that $Ax^2+By^2+Cz^2=1$ is the equation of a hyperboloid of two sheets that goes through the point $(1,-2,4)$ Working - I know that the equation for a hyperboloid of two ...
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1answer
36 views

Conical Equation

A quadric surface has the following equation: $2 x^2+3 y^2+3 z^2+J+16 x−18 y−6 z=0$ Enter a value of $J$ for which the quadric is : a) A single point b) The empty set Working - I honestly have ...
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1answer
31 views

I want to display a parabola defined by a line and a focus point but I don't get what is expected.

So I want to get the equation of the parabola from a line equation and a focus point. The line is defined by 2 points (x1,y1); (x2,y2) on a plane. and the focus point (fx,fy) is another point on the ...