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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length of a tangent

The two tangents to a circle are represented by $2x^2-3xy+y^2=0$ . A circle of radius=3 is in first quadrant . "A" is a point of tangency where one of these lines meet.What is length OA where $O$ is ...
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$y^2 = 2a(x+a\sin \frac{x}{a})$ and tangents parallel to $x$ axis

Prove that all the points on the curve $$y^2 = 2a(x+a\sin \frac{x}{a})$$ at which tangent is parallel to the axis of $x$, lie on a parabola. Here slope of tangent at $(h,k)$ must be $0$. After ...
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Proving that the line joining $(at_1^2,2at_1),(at_2^2,2at_2)$ passes through a fixed point based on given conditions on $t_1,t_2$

Problem:If $t_1$ and $t_2$ are roots of the equation $t^2+kt+1=0$ , where $k$ is an arbitrary constant. Then prove that the line joining the points $(at_1^2,2at_1),(at_2^2,2at_2)$ always passes ...
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69 views

Explicit formula for conformal map from ellipse to unit disc (interior to interior)

I was originally looking for a conformal map that maps a punctured unit disc to unit disc. The only answer I can find lead to this resource. The final step of the answer given rely on a conformal map ...
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32 views

tangent of an ellipsis $c^2=a^2m^2 + b^2$?

If $y=mx + c$ is a tangent to an ellipsis $(\frac{x^2}{a^2})+(\frac{y^2}{b^2})=1$ Show that $c^2=a^2m^2 + b^2$. So for this question, first off I tried to differentiate it using implicit ...
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3answers
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ellipse polar co-ordinate conversion

I have a somewhat trivial question out of interest. Given the equation of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ why is the substitution $x = \sqrt{a}\cos t$ and $y = \sqrt{b}\sin t$ ...
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What is the meaning of the locus of points P satisfying some conditions?

A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x axis and end point B lies on y axis. A point P(x,y) is taken on the rod in such a way that AP ...
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How to find the equation of circle that passes through ($5,3$) , ($7,-2$) and ($-4,4$) circle with center at origin ($0,0$) and radius $r$?

It is a challenge assignment on our class and I can't figure out how to solve it I always got stuck it is not the same as the other examples which are easy to solve. thanks
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What do I get if my Directrix and Focus are the same?

In my calculus class today we had a discussion about whether you'd get a vertical line or just a point when your directrix and focus become the same point. What would happen?
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Why is the focus of the parabola not within the parabola in the following result?

So i'm going through my book and try to solve the following question: Find the equation of the parabola which is symmetric about the y axis and passed through the point (2,-3). Since it passes ...
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4answers
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I do not get this question at all. I need to prove the an equation has a minimum. Quadratics involved.

Prove that $f(x)= (x-a)^2+(x-b)^2$ has a minimum when $x= \frac{a+b}{2}$. (Prove not verify) I do not get this question whatsoever, please help me.
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0answers
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angle between hrizontal and a line connecting the center of an oblate ellipse to a point in space

I would like to know how I can calculate the angle $\alpha$ in an oblate ellipse similarly to the sphere.
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1answer
60 views

Construct ellipse from two arbitrary points and a given focal point

Can an ellipse be constructed from these three given points: Focal point $\mathrm F$ Two arbitrary points $\mathrm U$, $\mathrm V$ lying on the ellipse The background is a orbital maneuver ...
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1answer
37 views

Conics and Loci Question (Hyperbolae and Circles)

A circle has the equation $x^2 + y^2 = r^2$. Tangents are drawn from a point $P(x_1,y_1)$ to the circle and these touch the circle at points $A$ and $B$. If the position of $P$ can vary and the locus ...
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2answers
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How to Change an equation into Ellipse Form

I know how to arrange a normal equation into an ellipse form, but this one is slightly different. $x^2+2xy+5y^2=1$ Any help with this would be greatly appreciated. Thanks
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General form to standard form regarding ellipse?

I've tried 2 hours to do this so I hope someone can help me: $$11400000=-0.64x^2+2560x-y^2+6000y$$ It says that it have to equal an ellipse with center at the point $(2000,3000)$ and a horizontal ...
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1answer
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Points with constant polar w.r.t to a tangent conic bundle

Consider the conic bundle of $\mathbb{P}^2(\mathbb R)$with matrix $$A(\lambda,\mu)=\begin{pmatrix} 0 & \mu & \mu \\ \mu & 0 & \lambda \\ \mu & \lambda & 0 \end{pmatrix}$$ This ...
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Is the Locus circle?

Locus of points such that sum of it's distances of them from four fixed points remains constant? Is the locus circle? I was not able to solve it as there were four radicals. Is it a theorem?
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A question an normal to the circle

The equation of the normal to the circle $(x-1)^2+(y-2)^2=4$ which is at a maximum distance from the point $(-1,-1)$ is (A) $x+2y=5$ (B) $2x+y=4$ (C) $3x+2y=7$ (D) $2x+3y=8$ Since its a normal to ...
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2answers
133 views

Intersection of cone and cylinder layout formula for sheet metal application

A common part in HVAC is a cylindrical pipe intersecting a truncated cone. I am designing a machine to mass produce this part. I would cut the parts out of sheet metal and roll them up to form the ...
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1answer
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How do you find $∠XPC$ + $∠XPB$ such that $PB+PC$ is maximum where $P$ is a point on $f(x) = (x-1)(x-3)(x-5)$?

Problem: $f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be ...
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How to derive the equation of tangent to an arbitrarily point on a ellipse?

Show that the equation of a tangent in a point $P\left(x_0, y_0\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, could be written as: $$\frac{xx_0}{a^2} + \frac{yy_0}{b^2} = 1$$ I've ...
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0answers
14 views

Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle

I'm doing some research on the Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle and I was wondering if anyone knew why we consider the integer lattice points within ...
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Imaginary tangents of parabola

For a parabola $y^2 = 4ax$ ,we can draw $2$ tangents from any point.If the point is outside of parabola then obviously we can draw $2$ tangents. If the point is on the parabola then the two tangents ...
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What simple topological properties of conic sections can be explored?

In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them? Edit: To clarify, ...
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Co-ordinate Parabola Circle Contained in it; Difference in maximum and minimum possible radius

If the Difference of radii of larget and smallest Circle passing through the focus of Parabola $$Y^2=4x$$ and toughing parabola in at least one point is My Approach Let Circle be $$C: (x-a)^2+y^2=r^2$$...
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Compute the shortest possible distance between a point on the graph of $f$ and a point on the graph of $f^{-1}$

Let $f(x)=(x+3)^2+\cfrac{9}{4}$ for $x\ge -3 $.Compute the shortest possible distance between a point on the graph of $f$ and a point on the graph of $f^{-1}$. My effort Let $P,Q$ be points on the ...
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GetThere Airlines currently charges $200$ dollars per ticket.How can they maximixe their revenue if they were to increase the price?

GetThere Airlines currently charges $200$ dollars per ticket,and sells $40,000$ tickets.For every $10$ dollars they increase the ticket price,they sell $1000$ fewer tickets. How much should ...
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2answers
114 views

How to determine the reflection point on an ellipse

Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point A is reflected by ...
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Partially differentiating the equation of a conic section

There was this question where a double degree equation of a conic section was given and the coordinates of the center of conic had to be found. The solution first partially differentiated the equation ...
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1answer
67 views

Find a parabola knowing its distance from a point.

I have the parametric parabola: $$ y=f(x)=C(x-4)(x-5)+D $$ where $D$ is fixed. I want to find for which value of $C$ the distance from the parabola to the point $(4,0)$ is exactly $\frac{1}{3}$ and ...
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33 views

No. of points determining a unique parabola

For a parabola, let Focus: $(a_1,b_1)$ Equation of directrix: $y-mx-c=0$ The equation of parabola is, $\sqrt{(x-a_1)^2+(y-b_1)^2}= \frac{|y-mx-c|}{\sqrt{1+m^2}}$ There are 4 parameters $m,c,...
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“Mean” ellipse inbetween two ellipses

I am dealing with two ellipses, described by bigger one: 30052069549920 - 560534420160 x + 3754285920 x^2 - 84631979520 y + 18247680 x y + 177708960 y^2 == 0 smaller one: -1431356032960 + ...
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2answers
118 views

Finding x-intercept of a parabola given one x-intercept

I am given an $x$-intercept of $-3-\sqrt{7}$ and I am asked to find the other intercept. I am having trouble since I don't have any other information but the given $x$-intercept. My guess is that the ...
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1answer
34 views

Vertex Form of Parabola - Why does it work?

Recently, I have been trying to plot parabolas of quadratic equations. First, I have to convert them to vertex form and then we can easily plot them. This makes me wonder why the vertex form of a ...
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1answer
58 views

Representing transformed ellipse

I am drawing ellipses using SVGs. An ellipse is described by center {x,y}, radiusX and radiusY. To be able to draw every ellipse, I also added rotate angle alpha. (As described here - every ellipse ...
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1answer
24 views

Tracing of a conic

I have my assignment of drawing a parabola with equation $y^2=16x$ . I cannot see how to do it. I cannot see any parameter to draw a parabola . One of my friends said use latus rectum but as I am a ...
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3answers
55 views

Conics (Ellipse): Complete the Equation to Give at least 1 point

The question asks: For which values of $a$ does the conic $4x^2+16x+5y^2-40y=a$ have at least one point? (State your answer in interval notation.) $a\in$ ___ I was able to understand that ...
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1answer
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Sketching a parametrised cone and a geodesic lying on it.

I just started a new module at University and I am having some trouble with parametrisation. I am given a parametrisation of a geodesic lying on a cone in notation $r(t)=x(t){\bf i}+y(t){\bf j}+z(t){\...
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Does a right circular cone only consists of pair of straight lines, hyperbolas, parabola, circles and ellipses? [closed]

I was reading about the conic sections and that a conic section includes pair of straight lines, ellipses, hyperbola, circles and parabola. Are all these 5 components enough to form a right circular ...
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1answer
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If an parabola has its focus at the (a,b) and has directrix at x=c…

If an parabola has its focus at the (a,b) and has directrix at x=c, what would the equation 4p(x – h) = (y – k)^2 look like in terms of a,b, and c?
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Eccentricity of a general ellipse

How to find the eccentricity of an ellipse $5x^2 + 5y^2 + 6xy = 8$ ?. I tried it by factorizing it into the distance form for a line and point but I failed. Please help
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What is the equation of the bottom half of the parabola $x + (y - 2)^2 = 0$?

A parabola has the equation: $$x + (y - 2)^2 = 0$$ I can't find the $y$ without getting the equation into some weird recursion.
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91 views

Parametrize an intersection of a plane and an elliptic paraboloid

I'm supposed to parametrize the intersection of the plane that has the equation $z = 5x + 3y$ and the 'elliptic paraboloid' with the equation $z = 3x^2+2xy+3y^2$ These two equations can also be ...
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Locus of vertex

A variable parabola of latus rectum $l $, touches a fixed equal parabola , the axes of the two curves being parallel . Then locus of the vertex of the moving curve is a parabola , then what is the ...
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57 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola [closed]

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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27 views

Point of intersection of tangents

If the distance of two points $P$ and $Q$ from the focus of of a parabola $y^2 =4ax$ are $4$ & $9$ then what is the distance of the point of intersection of tangents at $P$ and $Q$ from the focus....
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1answer
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How to change $ Cx^2 + Dy^2 + Ex + Fy + G = 0$ to$ (x-h)^2/a^2 ± (y-k)^2/b^2=1 $ using only the variables C, D, E, F, and G

Or, state the terms a,b,h,and k in terms of C, D, E, F, and/or G $Cx^2 + Dy^2 + Ex + Fy + G = 0$ $(x-h)^2/a^2 ± (y-k)^2/b^2=1$
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1answer
33 views

Distances in HP

A variable straight line passes through the fixed point $A(6,1)$ and meets the ellipse $x^2 + 2y^2 = 2$ at points $B$ and $C$. If $P$ is a point such that the lenghts $AB, AP, AC$ are in HP (harmonic ...
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2answers
21 views

Axes Rotation Problem

Given $$x^2 - 4xy + 5(\sqrt5y) + 4y^2 + 1 = 0$$ rotate the axes to eliminate the $xy$-term in the equation, then write the equation is standard form.