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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121 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
76 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
62 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
50 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
140 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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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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0answers
26 views

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
23 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
63 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
49 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
54 views

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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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
107 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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0answers
55 views

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
46 views

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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0answers
121 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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0answers
24 views

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
68 views

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

The polar equation $r = a \cos \theta$ represents which conic?
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32 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
33 views

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
42 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
57 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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1answer
71 views

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
149 views

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
177 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
108 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
45 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
90 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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0answers
55 views

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
30 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
86 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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1answer
31 views

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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2answers
138 views

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?
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1answer
57 views

Centroid of triangle formed by co-normal points

How can you prove that he centroid of a triangle formed by 3 co-normal points lies on the axis of the parabola?
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2answers
76 views

Orthogonal tangents to an ellipse [duplicate]

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
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1answer
70 views

The minimum distance from the circle $x^2+(y+6)^2=1$ to parabola $y^2=8x$?

What are the coordinates of the points on the parabola $y^2=8x$ which are at the minimum distance from the circle $x^2 + (y+6)^2=1$?
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1answer
35 views

Counting the dimension of a component of $\mathsf{hilb}^{2t+1}_{3}$

Consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, parametrizing varieties of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Consider the component $ ...
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1answer
79 views

Locate a point a given distance from another point on an ellipse

Similar to Point on circumference a given distance from another point, but for an ellipse. Unfortunately, the difference is non-trivial. I have an ellipse and a point (C) that is somewhere on the ...
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2answers
80 views

how do i write an equation in standard form by completing the square for $x^2 -9y^2-4x-18y=14$

I'm really having trouble with completeing squares i can solve for circles and ellipses but i can't seem to understand hyperbolas or parabolas, help would be deeply appreciated.
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1answer
30 views

Calculating an Ellipse given the Orbital Eccentricity and a Vertex?

I know that the formula for Eccentricity is e = c/a where c is the distance from the center to a focus and a is the distance from that focus to a vertex. I know the distance from the center of the ...
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1answer
47 views

Finding locus of centroid

Let AB be a chord of circle x^2 + y^2 = 3 which subtends 45 angle at P where P is any moving point on the circle. Then find the locus of centroid of triangle PAB Any help would be appreciated
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1answer
22 views

Finding Radical centre problem

Suppose 3 circles are drawn taking the 3 sides of a triangle as their diameters, what would be the radical centre of these circles? The options are circumcenter, orthocenter and incenter Any help ...
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2answers
172 views

Locus of centre of variable circle

I am not able to figure out this question What is the locus of the centre of a circle which touches a given line and passes through a given point, not lying on the given line? I think it's a ...
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1answer
62 views

Intersection of a 45 degree angle and an ellipse

If you are looking at the upper right quadrant of an ellipse centered at $(0,0)$, with $a=1$ and $b = 0.6$, and there is a $45$ degree line drawn from $(1, 0.6)$, how would I find the $(x,y)$ ...
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1answer
69 views

Identify the locus.

Let $A,B,C$ lie on a straight line. $B$ is lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The point of contact of the tangents from $A$ to these circles lies on ..... We ...
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1answer
35 views

If an ellipse has two radiuses, is there something like it, but with three or more radiuses?

If we say that a circle has one radius, and an ellipse has two, can I define figures that have three, four, or more radiuses? Also, how can I get that "radius"? In an ellipse that is 10 at its ...
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1answer
145 views

How to find the equation of a parabola with vertex on the line y = -3x?

Its axis are parallel to the y-axis and passing through (-7,13) and (5,1).
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2answers
58 views

Find the tangents to the following curve from the given point.

2x^2 + y^2 = 54 from (10,1) P.S. I still don't study calculus. This lesson is from analytic geometry and I have no idea how to solve it because my professor didn't teach it. So if someone could tell ...
3
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0answers
183 views

Equation of intersection of two cones

The equations of two cones are given; $(x-x_{0})^2+(y-y_{0})^2=\frac {(z-z_{0})^2}{m^2}$ and $(x-x_{1})^2+(y-y_{1})^2=\frac {(z-z_{1})^2}{m^2}$ How to find the equations of intersections 1) ...
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1answer
78 views

Find the equation of an ellipse

I have to find the equation of an ellipse which passes through the point $(3, 2)$, has center at the origin and major axis along the y-axis, i.e., is a vertical ellipse. No other info is given. I've ...