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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2answers
268 views

Find equation of the circular cross section of a unit sphere

I have a unit sphere in Cartesian coordinates: $x^2 + y^2 + z^2 = 1$ or in spherical coordinates: $x = \rho \sin(\phi) \cos(\theta)\\ y = \rho \sin(\phi) \sin(\theta)\\ z = \rho \cos(\phi)$ I ...
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1answer
98 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
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0answers
77 views

Please help in solving $ax^2 + bxy + cx + dy + e$ = 0

Sometime back when trying to work out how to solve $ax^2 - by^2 + cx - dy + e = 0$ I learned that the way to solve such forms is to 'square the terms' and give it the form $A^2 - B^2 - E = 0$, $A = ax ...
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0answers
125 views

Ellipses given focus and two points

I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane. There are several possible approaches but I'm not sure which is the best - both ...
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2answers
191 views

Tangent to Ellipse

I've been stuck on this problem for a while now. Not quite sure how to get at it. I've tried finding the derivative of the equation and using point slope form but cant get it to look like the defined ...
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1answer
159 views

Implicit derivitave of a general ellipse

Consider an ellipse centered at the point $(h,k)$. Find all points $P=(x,y)$ on the ellipse for which the tangent line at $P$ is perpendicular to the line through $P$ and $(h,k)$. I know the general ...
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0answers
43 views

Quadrature of the parabole

This exercise is from a course in mathematics history. Find U: S: V, where S is the area of ​​a parabolic segment, U is the area of the largest triangle that can fit inside the parabola segment ...
2
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1answer
116 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
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2answers
137 views

Intersection of two tangents on a parabola proof

There are two tangent lines on a parabola $x^2$. The $x$ values of where the tangent lines intersect with the parabola are $a$ and $b$ respectively. The point where the two tangent lines intersect has ...
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3answers
74 views

parametrise equation of a hyperbola

Any point on an ellipse can be wrttien as $(a\cos\theta,b\sin\theta)$, How could we genarilse this to a hyperbola?
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0answers
27 views

find inscribed ellipses in quads

I know in an convex quads,there are a family of inscribed ellipses. what I want to konw is when the semi-axis 'a' and four vertexs are given,how to determine the rotation angle.there may be three ...
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1answer
117 views

how to simplify a general plane conic section's equation by linear algebra?

When encountering a general plane conic section a11x^2+a12xy+a22y^2+b1x+b2y+c=0, i can write it in matrix form as a quadratic form of the vector [x,y,1]. by what then? what should be done to reach the ...
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0answers
191 views

Find angle given arc length and radius

I've got a function, $r(\theta)$, of the radius of an ellipse relative to one focus of the ellipse: $$ r(\theta) = \frac{l}{1 - e\cos \theta} $$ where $e$ is the eccentricity and $l$ is the semi-latus ...
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0answers
29 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
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0answers
74 views

Locus Problem .

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola is $y^2(2x+a)=a(3x+a)^2$
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2answers
60 views

Range of a parabolic shot

Prove that the range of a parabolic shoted from a height $h$, speed $v$ and angle $\alpha$ with the floor is maximum when $\alpha$ satisfies $\cos(2\alpha) = \frac{g h}{v^2 + gh}$ This is my ...
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6answers
223 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
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1answer
39 views

Ellipse features from either expanded form or general form

I have ellipses that are not aligned with the x-axis and are not centered at the origin. Hence, their defined by either of the following two equations: ...
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1answer
88 views

Deriving Rutherford Scattering Angle expression

OK, this was an assignment and I want to be sure I am on the right track here. It's math as much as physics. We want to find an expression for $\theta$, the Rutherford scattering angle. I have an ...
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0answers
25 views

Determine the equation of the directrix of the parabola 2y=(x-1)(x-3) and find the equations of the tangents to the curve ..

Determine the equation of the directrix of the parabola 2y=(x-1)(x-3) and find the equations of the tangents to the curve at the points where the parabola cuts the x-axis Can someone please help me ...
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1answer
60 views

hyperbolic orbits, deriving in cartesian coordinates

I was working on this and I wanted to be sure I wasn't too far off. Given: $\frac{\alpha}{r} = 1 + \epsilon \cos \theta$ where $\epsilon$ is eccentricity. Also $\frac{(x + x_0)^2}{A^2} = ...
2
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1answer
54 views

Product of the distance from foci to a tangent is a constant

I am supposed to determine what is the result of said product. Given $P(x_0,y_0)$, I need to calculate the distance from the foci to the tangent line that passes through $P$, and then multiply the ...
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1answer
124 views

Kepler, cartesian coordinates and ellipses

I am trying to see if I am on the right track with this. The problem: A kepler orbit (an ellipse) in Cartesian coordinates is: $$(1−\epsilon^2)x^2 + 2\alpha \epsilon x + y^2 = \alpha^2.$$ The task ...
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1answer
198 views

Expression for hyperbola on complex plane

The hyperbola $$x^2 - y^2 = 1$$ has a simple expression in the complex plane as $\{z^2 + \bar{z}^2 = 2\}$. Is there a similarly simple expression for a hyperbola ...
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1answer
51 views

What is the vertex of this parabola and it's min value?

Th equation of the parabola is $$2\left(x+\dfrac34\right)^2−\dfrac{25}8$$ What is the vertex and the min value? and do I just plug $x$ values into the equation to get the points on the graph?
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1answer
229 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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1answer
85 views

Common Normal Parabola Problem

Prove that two parabolas $y^2=4ax $ and $y^2=4c(x-b)$ cannot have a common normal other than the axis, unless $ b/a-c>2$. I couldn't think of a satisfactory approach. Please Help.
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1answer
45 views

How do I get the equation for this parabola in standard form?

How do I get the equation for this parabola in standard form? $ y = f(x)= 2x^2+3x-2$
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0answers
85 views

Polar equation of perimeter of half ellipse

x = Cx + a * cos(ang); y = Cy + b * sin(ang); Cx, Cy are cords of center. ...
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3answers
109 views

How to find the point on a parabola where x and y are equal?

On a parabola how could i find the point at which the y and x points are equal and meet on a point of the graph, algebraically?
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2answers
55 views

ellipse boundary after rotation

Assume I have this vertical ellipse with a certain major axis $a$ and minor axis $b$. If we take the center of the ellipse to be at $(0,0)$, then the top right small red circle will be at $(b,a)$. ...
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2answers
61 views

Proving a parabola property

I need help with this question that i attempted to solve using the equation $y^2=4ax$ : "Prove that on the axis of any parabola there is a certain point which has the property that,if a chord PQ of ...
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1answer
2k views

How to derive the equation of a parabola given a focus and a directrix not parallel to the x or y axis?

I was wondering if it is possible to derive a general form of a parabola given any focus and directrix. So far all the materials I have come across only show the derivation for a parabola equation ...
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2answers
80 views

Finding centre of ellipse using a tangent line?

I need to determine the centre coordinates (a, b) of the ellipse given by the equation: $$\dfrac{(x-a)^2}{9} + \dfrac{(y-b)^2}{16} = 1$$ A tangent with the equation $y = 1 - x$ passes by the ...
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1answer
79 views

Parallel Curves to a Parabola

I have been modelling parallel curves to a parabola and realise if the parallel curve to a parabola is offset enough then the curve will overlap. I came across this research paper to explain why a ...
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1answer
24 views

Polar of a point locus of the point?

I'm trying to solve this problem but can't understand what is meant by this "polar" The question is as follows., "If the polar of any point with respect to the parabola $y^2=4ax$ touches the circle ...
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1answer
172 views

Locus of a point where two normals meet?

Another exam question, "Find the locus of a the point such that two of the normals drawn through it to the parabola $y^2=4ax$ are perpendicular to each other." Does the locus mean the point of ...
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2answers
238 views

Prove the straight line as a tangent to a parabola

I was going through some past exam papers and I came across this problem and I'm bit puzzled on how to approach this, could someone please help me out? Equation of the parabola $y^2-7x-8y+14=0$, ...
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0answers
95 views

rotating parabolas in 3D to get part of a circle

Say you have a unit-circle with its center at (0,0), and you "cut out" the upper-right quadrant. You rotate this segment around the Y-axis and the orthographic projection is the upper-right segment of ...
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3answers
70 views

Finding the second radius of an ellipse given the first radius and the center

I know the coordinates of A, B and C. A and B are on the axis L1. From that information, I can find the coordinates of the center, the length of radius r1 and the equation of L1 (see picture). Then I ...
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1answer
148 views

Determining the direct and transverse tangent lines for two non-overlapping ellipses

I am trying to determine the direct and transverse lines for two non-overlapping ellipses. I specifically mean that the two ellipses are totally separated from each other with no shared regions. I ...
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2answers
119 views

pick point on parabola so 2 conditions are true

You have a parabola $$y=ax^2+bx+c $$ We know $a,b$ & $c$. On this parabola you have to pick a point A where the following conditions are true: 1) If you draw a tangent line in this ...
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1answer
212 views

Hyperbola in polar coordinates, what's wrong?

I read that the equation of a conic in polar coordinates is $$r=\frac{l}{1+e\cos \theta}.$$ But when I try to reduce the hyperbola $$x^2 - y^2 =1$$ to that form by setting $x=r\cos \theta $, $y=r ...
3
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0answers
51 views

Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
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1answer
23 views

Rewrite a west to east parabola in standard form

$$8y^2+96y-12x+240=0$$ I'm not sure how to approach that problem because there's a $\frac23y^2$ to deal with
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1answer
155 views

Ellipse fun (given an arc length, calculate theta)

I can see the general solution.. but I was wondering if the wizards here could help me along. :) Given an ellipse with a known major and minor axis. Take a known r(T) such as, the major or minor ...
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0answers
40 views

Why loci of conic sections are defined in the way they are?

I understand how conic sections are produced i.e. when a plane cuts a double nappe right circular cone at different angles, we get different types of conic sections like parabola, ellipse etc. But I ...
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1answer
65 views

Graphing Hyperbolas

I know that a Hyperbola is in the form of: $$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$$ But how would I graph it? I know that a Hyperbola has two asymptotes that the graph gets infinitely close to ...
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0answers
91 views

Plotting an elliptical arc given 3 points, radius ratio and angle

I'm trying to plot an elliptical arc. I know the starting point $P_1$, ending point $P_2$ and a control point $P_3$. I'm also given the ratio of radii $a/b$ and the angle $\theta$ of the ellipse. As ...
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0answers
89 views

Dome with a parabolic shape?

Here is the question: The dome over a town hall has a parabolic shape. The dome measures 48 m across and rises 12 m at its centre. a) Determine the quadratic equation that models the shape of the ...