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.

learn more… | top users | synonyms (3)

0
votes
0answers
16 views

Universal Parabolic Constant

I recently learned of a constant that arises in parabolas, similar to that of $\pi$ for circles. Like $\pi$ being the ratio of the circumference of the circle to its diameter, this constant ...
0
votes
1answer
42 views

Relation between $a$ and $b$ when equation of obtuse angle bisector is $ax+by-3=0$

The combined equation of bisector of angles between the lines $L_1$ and $L_2$ is $$2x^2-3xy-2y^2-x+7y-3=0$$ $P(4,-3)$ is a point on $L_1$. If the equation of obtuse angle bisector is ...
0
votes
1answer
65 views

How to prove that the ellipse is a periodic orbit knowing that the orbital derivative of a function V is zero on there

The question is as follows: Show that the orbital derivative of the function $V=(1-x^2-2y^2)^2$ is zero on the ellipse $x^2+2y^2=1$, and explain why you can deduce that the ellipse is a periodic ...
1
vote
1answer
39 views

Determining the normal of an ellipse

Given I have (in a 2D coordinate system) an ellipse with the center at $(c_x,c_y) = (0,0)$ where I do not know the actual value of the major an minor axis but I have the ratio $r=\frac{a}{b}$ and an ...
4
votes
1answer
43 views

Is there a name for this quantity which is similar to the focus of a parabola?

Suppose we have parabola $y=ax^2+bx+c$, which has focus at $(-\frac b{2a},\frac 1{4a}-\frac {b^2}{4a}+c)$. There is a line $\ell$ at $y=\frac{a^2-b^2}{4a}+c$ which has the following property: any ...
0
votes
1answer
37 views

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 ...
1
vote
1answer
16 views

$y^2 = 2a(x+a\sin \frac{x}{a})$ 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 ...
2
votes
2answers
100 views

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 ...
0
votes
1answer
52 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 ...
0
votes
2answers
30 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 ...
0
votes
3answers
25 views

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 $y = \sqrt{b}\sin t$ ...
0
votes
2answers
32 views

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 ...
-1
votes
2answers
44 views

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
0
votes
1answer
41 views

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?
1
vote
3answers
38 views

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 ...
0
votes
4answers
62 views

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.
1
vote
0answers
17 views

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.
1
vote
1answer
50 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 ...
0
votes
1answer
34 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 ...
0
votes
2answers
22 views

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
0
votes
1answer
20 views

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 ...
0
votes
1answer
16 views

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 ...
0
votes
1answer
41 views

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?
0
votes
2answers
38 views

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 ...
2
votes
2answers
99 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 ...
4
votes
1answer
32 views

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 ...
2
votes
3answers
51 views

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: $$\frac{xx_0}{a^2} + \frac{yy_0}{b^2} = 1$$ ...
0
votes
0answers
13 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 ...
1
vote
0answers
28 views

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 ...
1
vote
0answers
38 views

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, ...
0
votes
0answers
40 views

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: ...
0
votes
1answer
37 views

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 ...
0
votes
1answer
32 views

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 ...
3
votes
2answers
104 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 ...
0
votes
0answers
27 views

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 ...
0
votes
1answer
61 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
22 views

“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 + ...
0
votes
2answers
75 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 ...
1
vote
1answer
33 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 ...
1
vote
1answer
56 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 ...
1
vote
1answer
22 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 ...
1
vote
3answers
47 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 ...
0
votes
1answer
31 views

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 ...
1
vote
1answer
24 views

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?
1
vote
2answers
22 views

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
1
vote
1answer
43 views

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.
1
vote
1answer
81 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 ...
0
votes
1answer
30 views

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 ...
1
vote
2answers
56 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.