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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xy points in the perimeter of a rotated ellipse

How can I calculate the $(x,y)$ position of every point on the perimeter of a rotated ellipse? I have found the equations for a non-rotated ellipse $x=a \cosθ$ $y=b \sinθ$ What are the formulas if ...
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33 views

maximum radius of a circle inscribed in an ellipse

Consider an ellipse with major and minor axes of length 10 and 8 resp. The radius of the largest circle that can be inscribed in this ellipse, given that the centre of this circle is one of the focus ...
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107 views

How to plot the graph of parabola?

I am new in the conics so there are two confusions: 1. How to plot the graph of this parabola equation $y^2-x-2y-1=0$? 2. what would be the equation of parabola when vertex $(3,2)$ and ends of focal ...
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Practise Calc Question

How is the radius $10$ in the circle equation: $x^2+y^2+6x-4y+3=0$? My work: Standard Form for circle: $(x-a)^2+(y-b)^2=r^2$ $X:$ $X^2+6x+[?]=-3+[?]$ $(x+3)^2= X^2+6x+9=-3+9 X^2+6x+9=6$ $Y:$ ...
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It's about Parabolas, I just can't seem to solve it…

Write the equation in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. $$x+y^2-8y=-20$$ I have seen some students answer it, but I just don't ...
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18 views

Show that the foci of $(-8t, 4t^2)$ for varying values of $t$ is a parabola.

Show that the foci of $(-8t, 4t^2)$ for varying values of $t$ is a parabola. Also, how to attempt loci questions in general?
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1answer
29 views

What is the approximation equation for making the day/night wave

Basically, I have a program that will graph the day/night shade similar to this page: http://www.timeanddate.com/worldclock/sunearth.html Could any of you give me the equation for graphing a line ...
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Locus of centre of circle in Lambert theorem

A beautiful theorem, when three tangents to a parabola form a triangle,the focus of the parabola lies on the circumcircle of the triangle. But what is the locus of the centre of the circumcircle of ...
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6answers
77 views

What is one way to prove that there exists no ellipse that matches the exact curvature of the sin wave?

Preferably by not graphing both and showing they don't match visually. By the sin wave, I mean just plain old y=sinx.
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1answer
22 views

Only one normal can pass through the focus of the parabola $x^2=4ay$

I've been having trouble with the following question: Show that only one normal can pass through the focus of the parabola $x^2=4ay$ and find from which point on the parabola it originates. ...
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2answers
19 views

Solve system of two conic equation

I need to solve three types of system of equations in general form: System of two linear equation ($Ax + By + C = 0$) which can be done perfectly by calculating D, Dx, Dy. System of two equations ...
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1answer
41 views

Equation of hyperbola

What is the equation of hyperbola if all axes (transverse axis, conjugate axis, principal axis) are along the coordinate axis (x and y axis), and passing through the point $(-3,4)$ and $(5,6)$. I ...
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6answers
65 views

Finding the largest coordinate of $y$ given the ellipse $x^2+y^2-xy=1.$

The equation of the ellipse is $x^2+y^2-xy=1$. I am asked to find the the coordinate of the point $p$ on the ellipse with largest $y$ value. Thanks in advance! So I have that ...
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1answer
37 views

Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$

Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$ So far I have said: As the circles have centre $(0,0)$ their equations are of the form ...
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1answer
27 views

Standard form of hyperbola

I've tried to solve this question, but I'm stuck. The question is: Find the equation in standard form for the hyperbola centered at $(5,5)$ with one focus at $(-7,5)$ and eccentricity at ...
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2answers
39 views

Eliminate xy term of conic

For the following problem, I am trying to eliminate xy, and I've tried numerous times to solve if with no luck. I need to find the general form of the equation rotated to eliminate the xy term. $ ...
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1answer
85 views

Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...
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2answers
65 views

Find the equation of the hyperbola with a given foci and a transverse axis

I know this is a homework but then I need to know how to solve this stuff. Just this one question will do to have a reference to answer the other questions that are like this. Please teach me the ...
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2answers
61 views

Plotting an Ellipse after an Ellipse Fit

I wonder if someone can assist my understanding as I'm a bit stumped with this... I have taken the following (x,y) data which lies roughly on an ellipse: $$ \begin{pmatrix} 0.000234491 & 6855810 ...
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2answers
45 views

How to construct the point of intersection of a line and a parabola whose focus and directrix are known?

I found this problem in Polya's "How to solve it". It goes as follows Using only a straight edge and a compass, construct the point(s) of intersection of a given line and a parabola whose focus ...
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1answer
42 views

Calculating tangent on ellipse

I want to calculate the slope of the tangent at one point of an ellipse whose centre is shifted towards the coordinates $(x_c;y_c)$ and also rotated by an angle $\alpha$ around its centre. Now, I have ...
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2answers
125 views

Find the length of the longest line segment contained in the given region

Consider the region represented by the following in the $x-y$ plane. $y=v$, $x=u+v$ and $u^2+v^2\leq1$ $u$ and $v$ are parameters. What is the length of the longest segment contained in the given ...
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2answers
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HELP: find the type of a conic from the given equation

However, I am not sure what conic type it is. Should it be divided by 4 in order to get a standard form of a hypebola? Any help will be appreciated. Thank you!
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Is there a Focal Point/Area/Line of a Parabola for not perpendicular Lines

I'm not sure if this is mathematical enough for this forum, since it's my first post, but please don't be too harsh! So my question is: If the incoming lines of a Parabola come in perpendicular to ...
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1answer
21 views

Ellipse(Finding the center, vertices)

So this is the equation 16x^2 + 9y^2 = 144 So this is what I did: 16x^2/144 + 9y^2/144 = 144/144 x^2/9 + y^2/16 = 1 a^2= 9 ; a = 3 b^2= 16 ; b=4 so if I solve for the c c^2 = a^2 - b^2 c^2 = ...
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1answer
53 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
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1answer
47 views

How do i find the equation of a parabola given the max and two points

The points of the parabola are (10,0) and (42,0). The maximum is 22. If you could show me the equation and how to find it, it would be greatly appreciated.
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4answers
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Equation of a Circle which share the same center

How to find the equation of the circle which passes through the point $(-2,-4)$ and has the same center as the circle whose equation is $x^2+ y^2 -4x - 6y -23$ ?
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4answers
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Parabola is an ellipse, but with one focal point at infinity

While I was reading about conic sections, I came across the following statement: A parabola is an ellipse, but with one focal point at infinity. But it is not clear to me. Can someone explain it ...
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1answer
64 views

Maximum/Minimum of Curvature - Ellipse

Find the sum of the maximum and minimum of the curvature of the ellipse: $9(x-1)^2 + y^2 = 9$. Hint( Use the parametrization $x(t) = 1 + cos(t)$) Tried to use parametrization like that, but then ...
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Foci of ellipse and distance c from center question?

I don't understand how you would figure out an exact formula for the linear eccentricity (distance from the center to either focus) $c$ of an ellipse, being $c^2=a^2-b^2$, where $a$ is the length of ...
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0answers
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Equation of the locus of the foot of perpendicular from any focus upon any tangent to the ellipse ${x^2\over a^2}+{y^2\over b^2}=1$

Find the equation of the locus of the foot of perpendicular from any focus upon any tangent to the ellipse ${x^2\over a^2}+{y^2\over b^2}=1$. will it also be an ellipse?
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63 views

Intersection between sphere and ellipsoid

I am failing since two days to compute and to plot the intersection of an ellipsoid in parametric notation ...
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Cubic curves vs conics

What is the main difference between cubic curves and conics, i.e. why can cubic curves develop singularies while conics cannot? Is this in some way related to Poincare-Bendixon theorem of chaos ...
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1answer
57 views

Which conic is represented by $r = a \cos \theta$

The polar equation $r = a \cos \theta$ represents which conic?
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31 views

Problem on co-ordinate geometry

Suppose the circle with equation $x^2 + y^2 + 2fx + 2gy + c = 0$ cuts the parabola $y^2 = 4ax$, ($a > 0$) at four distinct points. If d denotes the sum of ordinates of these four points, then find ...
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1answer
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Finding the focus point of a conic with equation $ay^2 + bx = 0$.

A conic has equation $$ay^2+bx=0$$ where $a=5$ and $b=-315$. If the focus point is at $(F, 0)$ then what is the value of $F$ to 2 decimal places? Hi, I want to check if i have applied the correct ...
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1answer
40 views

Equation for focus and directrix

Is it possible to get a focus and directrix straight from the equation itself or through a formula? For example, in $y = (x-2)^2 + 1$, you can tell from the equation that the vertex is $(2,1)$. Or ...
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2answers
50 views

Hyperbolas - Standard Form

This is probably a simple question but if $y = \frac{1}{x}$ is a hyperbola, then how does it comply with the standard form of a hyperbola?
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ellipse circumference

Here is a Wikipedia article about the circumference of an ellipse: http://en.wikipedia.org/wiki/Ellipse#Circumference I don't know how Ramanujan developed the following approximation for the ...
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2answers
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Ellipse Diagonal's Length/Equation [closed]

Excuse the vagueness of this question, but how can you find the equation and distance for the diagonal of any given ellipse, that is, the line from the most-northwestern point to the most southeastern ...
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3answers
145 views

Focus of parabola with two tangents

A parabola touches x-axis at $(1,0)$ and $y=x$ at $(1,1)$. Find its focus. My attempt : All I can say is that as angle subtended by this chord at focus is $90^\circ$ as angle between tangents is ...
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1answer
90 views

Rotation of conics sections using linear algebra

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then ...
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1answer
35 views

Finding perimeter of an ellipse accurately

How could you accurately find the perimeter of an ellipse accurately? This formula: $$p\approx 2\pi\sqrt{\dfrac{a^2+b^2}{2}}$$ (Where 'a' is the distance from the center of the ellipse to the ...
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4answers
70 views

Interesting association between tangent lines of slope one and ellipses

Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse? ...
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Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
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1answer
26 views

How to identify any point inside or outside the given cone?

The equation of a double circular cone with a vertex $p=(a,b,c)$ with the generating angle $t$ is given by $(x-a)^2+(y-b)^2= \frac{(z-c)^2}{t^2}$ How do I identify the point ...
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1answer
56 views

How to find centre,vertics,foci,focal radii,letus rectum… when exists of a general quadratic equation in x and y

Is there a generalized way( a particular conic section of any shape,for instance an ellipse without determining its major/minor axis) to find the centre,vertics,focus,focal radii,letus ...
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need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
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Area of triangle inscribed in a parabola

How can u prove that the area of the triangle inscribed in a parabola is twice the area of the triangle formed by the tangents at the vertices?