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)

1
vote
1answer
36 views

Find a specific rectangle in an ellipse

For a software developpment, I need to find a rectangle that fits in an ellipse. I have an outer rectangle (left, top, width and height) and a function that draws an ellipse in it. Now I need to know ...
0
votes
1answer
13 views

finding parabola equation by angle and 2 points [closed]

ok so I got a kind of mechanic question. I got two point (0,0) and (8,0) (8 meters between), I got an angle at X=0 of 20 degree and I got deaccelaration of 10 m/sec^2. how can I find the max point of ...
2
votes
1answer
35 views

Rolling ellipse on line - tangent and normal of roulette

Suppose that an ellipse is rolling along a line. If we follow the path of one of the foci of the ellipse as it rolls, then this path formes a curve - namely an undulary. Now consider the following ...
0
votes
1answer
18 views

Grade 10 Quadratic equation

This was on my year 10 maths test and I gave up with 40 mins to complete: Basically you were given the coordinates: y intercept : (0,10) 1 x intercept: (10,0) and y value of the vertex: +15 Can ...
17
votes
9answers
3k views

Why is an ellipse, hyperbola, and circle not a function?

I am aware of the vertical line test. If you place a vertical line over a shape, and if it crosses more than once, it fails the vertical line test and is no longer a function. But I don't understand ...
0
votes
0answers
25 views

Show a curve has no factor of degree 1 or 2

I have to show that $ h(x,y)=y^{2}(x^{2}+x+1)-x^{2} $ has no factors of degree 1 or 2. I know that h contains infinitely many points and is singular at the points (1,0,0), (0,1,0) and (0,0,1). I am ...
0
votes
1answer
19 views

How to convert formulas for different standard parabolas?

There are 4 types of standard parabolas , and I'm supposed to remember many formulas about them like tangent , normal etc. But the problem is , if i know a certain formula for $y^2=4ax $ how can i ...
0
votes
1answer
19 views

help needed in understanding general conics proof

The origin is a centre of a general conic of second degree iff the coefficients of linear terms vanish. $ (\Rightarrow)$ part: Let $$ Q(x,y)\equiv ax^{2}+2h xy+ by^{2} + 2gx+2fy+c=0$$ books ...
1
vote
1answer
46 views

Arc length of parabola between two points

Well lets take a parabola of the equation $y = f(x)$ where $f(x)$ is obviously a $2^{nd}$ degree function. Now lets take two points at $x=a$ and $x=b$ . So can anyone please help me to find that ...
0
votes
1answer
75 views

Focus of a parabola, without derivatives

I have a seemingly easy question, but I have no clue how to find out its answer. I have the function $$f(x)=\tfrac{1}{8} x^2$$ This function is for (a parabolic cross-section through) a paraboloid ...
0
votes
1answer
17 views

transformation of conic section

Given is the conic section $x^2 +xy + y^2 +2x +3y -3 = 0$. The following tasks: 1.) What is the coordinate matrix $A_1 = M_{\beta} (\sigma) $ of the bilinearform? 2.) do the transformation and ...
4
votes
2answers
52 views

Better substitution calculating integral?

I'm calculating $$ \iint\limits_S \, \left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}} \right)^\frac{1}{2} \, dA$$ with $$S =\left\{ (x, \, y) \in \mathbb{R}^2 : ...
2
votes
1answer
30 views

Given area of sector and a starting angle from focus of an ellipse, finding angle needed to get area.

Problem Background: I'm trying to make a rough simulation of Kepler's second law (equal areas over equal time) and to do this I've divided the area of the ellipse into some number of pieces. I want ...
-1
votes
1answer
35 views

Conic involving circle question. [duplicate]

The question is: If the curves $ax^2+4xy+2y^2+x+y+5=0$ and $ax^2+6xy+5y^2+2x+3y+8=0$ intersect at four concyclic points then the value of a is???? The options are: a) 4 b) -4 c) 6 d) -6 I've ...
2
votes
1answer
23 views

Graphing Circles, Ellipses, Parabolas, and Hyperbolas

I need help plotting a curve on a graph where the distance from focus1 is always the same ratio to the distance from focus2. For instance, lets assume focus1 is -5 along the x axis, and focus2 is +5 ...
3
votes
5answers
77 views

Find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$

How to find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$ Let $y^2-xy-2x^2 =0...(1)$ and $y^2=x-2...(2)$ In equation (1) coefficient of $x^2 =-2; y^2=1, 2xy =\frac{-1}{2}$ ...
3
votes
3answers
88 views

Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline?

Given the ellipse $$3x^2-x+6xy-3y+5y^2=0$$ find the following: semi-major axis, $a$ semi-minor axis, $b$ displacement of centre from origin (or coordinates of centre of ellipse $(h,k)$) angle of ...
1
vote
3answers
32 views

Find equation of tangents to hyperbola

$$\frac{x^2}{4} - \frac {y^2}{16} = 1$$ There is a point $(1,2)$ where $2$ lines pass through and are a tangent to both curves. How do I find the equation of both lines?
0
votes
1answer
25 views

Is it possible to calculate the volume of a parabolic arch?

Given that you know the equation of a parabola that only has positive values, is it possible to find the volume of the parabolic arch itself? NOT the volume of space underneath the arch. I asking ...
1
vote
1answer
26 views

Find the area of triangle APB, where P is a point $(a\cos\theta, b\sin\theta)$ on an ellipse and $A, B$ are its radii points $(a,0) (0,b)$

A point $P(a\cos\theta, b\sin\theta)$ sits on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The points $A$ and $B$ have coordinates $(a,0)$ and $(0,b)$ respectively. Show that the area of ...
0
votes
1answer
27 views

Let $P_1 = (x_1, y_1)$. Describe $P = (x,y)$ in $\mathbb{R}^2$ s.t. $||P-P_1|| = 9$ by identifying conic and finding its equation

Let $P_1 = (x_1, y_1)$. Describe the set of all points $P = (x,y) \in \mathbb{R}^2$ such that $||P-P_1|| = 9$ by identifying the type of conic and finding its equation. I'm sorry, but this question ...
0
votes
2answers
32 views

Find the equation of the locus of the mid-point between an elliptical point and its directrix

I'm struggling with this question: The point $P$ lies on the ellipse $x^2+4y^2=1$ and $N$ is the foot of the perpendicular from $P$ to the line $x=2$. Find the equation of the locus of the ...
0
votes
0answers
34 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
1
vote
0answers
16 views

Hyperbola problem : From a point P(1,2) pair of tangents are drawn to a hyperbola H in which one tangent to each arm of hyperbola…

Problem : From a point P(1,2) pair of tangents are drawn to a hyperbola H in which one tangent to each arm of hyperbola. Equation of asymptotes of hyperbola H are $\sqrt{3}x -y+5=0$ and ...
1
vote
0answers
20 views

Trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ takes on different shapes.

Note: This is a homework problem. I'm trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ is an ellipse or the empty set if $ac-b^2\gt 0$. There are others to show but if I can understand this ...
3
votes
0answers
26 views

Find the minimum radius of the circle which is orthogonal to two given circles

Problem : Find the minimum radius of the circle which is orthogonal to both the circles $x^2+y^2-12x+35=0$ and $x^2+y^2+4x+3=0$ . Solution : Let the equations : $x^2+y^2-12x+35=0.....(i)$ and ...
0
votes
1answer
50 views

Find the equation of the parabola with its vertex on the line $2y-3x=0$?

Its axis of symmetry is parallel to the x-axis, and it passes through the two points $(3,5)$ and $(6,-1)$
1
vote
2answers
35 views

If the length of tangent drawn from an external point P to the circle of radius $r$ is $l$ , then prove that area…

Problem : If the length of tangent drawn from an external point P to the circle of radius $r$ is $l$ , then prove that area of triangle form by pair of tangent and its chord of contact is ...
1
vote
1answer
71 views

Defining ellipse using points and normal vectors from them

There is an article on how to detect circles in images using pairs of gradient vectors (assuming the circle is dark and background is bright). The thing is that gradient of image intensity at each ...
0
votes
0answers
57 views

Asymmetric hyperbola-type curve? (for fitting to data)

I have this question: what would be the name and equation of a curve which resembles a parabola but has not the requirement of symmetry? So the general parabola equation is: $y=ax^2+bx+c$ I must ...
0
votes
1answer
17 views

Finding angle of a spigot that produces a parabolic fountain of water

I am currently doing a math exploration and I need help in determining how to find the angle of a spigot that would maximize the area under a parabolic fountain. I thought of this topic to ...
0
votes
2answers
102 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
0
votes
1answer
35 views

Conic sections in standard form

I'm trying to convert the equation $$x^2 +2y^2 +4x-4y+4=0$$ into its standard form by choosing a new set of axes. Yet, when I go down the conventional route, there is no xy term so ...
1
vote
1answer
42 views

a problem with Stokes' theorem(curl)

If L is the circle which you get from the intersection between the sphere $$ x^2+y^2+z^2=1, y=x\sqrt(3) $$ and $$ I= \int_L (y-z)dx+(z-x)dy+(x-y)dz $$ so |I| equals to? but i dont understand how the ...
0
votes
1answer
19 views

Parabola max. number

If the directrix and the tangent at vertex of a parabola are given then what is the maximum number of parabolas that can be drawn? Well according to me the answer should be 1 because the distance ...
1
vote
1answer
107 views

Prerequisites for Appolonius Conics?

I want to get Thomas Heath's version of Apollonius's Conic Sections. Does anyone know the prerequisites to understand everything in this book? I heard I would need the Euclid's Elements book on Solid ...
0
votes
0answers
40 views

Roulette of a parabola - Delaunay-Surface

I've problems to understand an equation I've found in various books and papers. Maybe someone could help me and explain it a little bit more precisely. I colored the equation in yellow (the picture ...
2
votes
0answers
33 views

Find the number of points on ellipse $\frac{x^2}{50}+\frac{y^2}{20}=1$ from which pair of perpendicular…

Find the number of points on ellipse $\frac{x^2}{50}+\frac{y^2}{20}=1$ from which pair of perpendicular tangents are drawn to ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ Normal from a point ...
4
votes
3answers
35 views

If A represents area of ellipse $3x^2+4xy +3y^2=1$ then the value of $\frac{3\sqrt{3}}{\pi}A = ?$

If A represents area of ellipse $3x^2+4xy +3y^2=1$ then the value of $\frac{3\sqrt{3}}{\pi}A = ?$ My approach : Since area of ellipse is $\pi ab$ where a is semi major axis and b is semi minor ...
1
vote
2answers
23 views

Ellipse 1 : E$_1 = \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ( a > b)$ another ellipse 2 : $E_2$ which passes thru…

Problem : Ellipse 1 : E$_1 = \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ( a > b)$ another ellipse 2 : $E_2$ which passes thru extremities of major axis of $E_1$ and has its foci at ends of its minor axis. ...
1
vote
1answer
10 views

Equation of a parabola with vertex $V$ and point $P$

Find the equation of the parabola which has the given vertex $V$, which passes through the given point $P$, and which has the specified axis of symmetry. $V(4,-2), P(2,14)$, vertical axis of ...
0
votes
3answers
38 views

How to determine family of circles passing through two given points?

The question asks to show that the equation of any circle passing through two given points takes a certain form. I have obtained the points as being $(2,1)$ and $(2,-1)$ but I'm not sure as to how to ...
0
votes
2answers
84 views

Find all natural number solutions to: $20x^2 + 11y^2 = 2011$

I believe that the equation $20x^2 + 11y^2 = 2011$ describes an ellipse. I don't know how to solve for the $x,y \in \mathbb{N}$ that satisfy this equation.
6
votes
1answer
64 views

shape created by parabola

What would be the name of the shape that is the set of all points such that they are equidistant from the point $(0,1)$ and to the parabola $y=x^2$. Here is a desmos graph that generates the ...
1
vote
1answer
37 views

Find the number of common tangents to $y^2=2012x$…

Problem : Find the number of common tangents to $y^2=2012x$ and $xy =(2013)^2$ Solution : Common tangent will have slope equal to both curves. therefore, differentiation both the curves we get ...
1
vote
0answers
15 views

P is a point t on the parabola $y^2=4ax$ and PQ is a focal chord. PT is a tangent at P and QN is a

Question : P is a point t on the parabola $y^2=4ax$ and PQ is a focal chord. PT is a tangent at P and QN is a normal at Q. Find the minimum distance between PT and QN. Solution : Since the ...
0
votes
2answers
53 views

An equation of a tangent to the parabola $y^2=8x$ is y=x+2. the point on this line from which the other tangent

Problem : An equation of a tangent to the parabola $y^2=8x$ is y=x+2. the point on this line from which the other tangent to the parabola is perpendicular to the given tangent is given by ... ...
0
votes
2answers
70 views

The point of intersection of the tangents to the parabola $y^2=4x$ at the points where the circle $(x-3)^2+y^2=9$

Problem : The point of intersection of the tangents to the parabola $y^2=4x$ at the points where the circle $(x-3)^2+y^2=9$ meets the parabola, other than the origin, is .. Solution : Point of ...
0
votes
1answer
24 views

The tangent at the point $P(x_1,y_1)$ to the parabola $y^2=4ax$ meets the parabola $y^2=4a(x+b)$ at Q and R,

Problem : The tangent at the point $P(x_1,y_1)$ to the parabola $y^2=4ax$ meets the parabola $y^2=4a(x+b)$ at Q and R, the coordinates the mid point of QR are ? Solution : Tangent from a point ...
0
votes
1answer
85 views

A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and int…

Problem : A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and intersects its axis at Q. If angle SPQ is $\frac{\pi}{2}$ find the equation of the ...