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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5
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6answers
207 views

Area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$.

Find the area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$. First I saw that this cannot be a circle ($xy$ term), and it cannot be an ellipse with axes parallel to the coordinate axes. But ...
4
votes
2answers
62 views

Fast method to find the tangent line to a conic section: why does it work?

My teacher taught me this fast method to determine the equation of the tangent line to a conic section. In the Netherlands this is called "eerlijk delen" or literally translated into English "fair ...
-3
votes
2answers
47 views

Water Density and Fluid Force (question below) [closed]

I've been trying to study the question and the answer below. Can someone tell me how to start this problem myself? I don't understand why they named one fourth of the circle equation the whole ...
3
votes
1answer
56 views

How homogenization of line and curve works?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
2
votes
1answer
32 views

Parabola properties assumptions

I am trying to model projectile trajectory but I'm having some trouble. I didn't realise parabolas are this complicated... I have some assumptions that I would like to be clarified. If I specify a ...
3
votes
2answers
80 views

Show that $PF.PG=b^2$ in a hyperbola

If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then ...
3
votes
3answers
118 views

Property of ellipses involving normals at the endpoints of a focal chord and the midpoint of that chord

While solving a book on ellipses, I came across the following property of an ellipse which was given without proof :- If the normals be drawn at the extremities of a focal chord of an ellipse, a ...
0
votes
0answers
17 views

Measure best fitting major and minor axis length given 3 points on an ellipse

I am trying to measure the parameters of an ellipse in an image. I have the center, the rotation of the ellipse. I am trying to find the best fitting major and minor axis length based on 3 given ...
0
votes
2answers
51 views

Parabola describing projectile motion.

I am trying to create a function that will generate a parabola that describes projectile motion. Here are my inputs: The starting x-y coordinate of the throw The initial x-y velocity vector. I ...
0
votes
1answer
21 views

Test if a point is within 2 parametric “cut-off” ellipses

I have 2 parametric ellipses, both represented using the standard parametric equation of an ellipse: $$x = h + a \cos t $$ $$y = k + b \sin t $$ Lets say that the ellipses are cut-off at (see ...
0
votes
1answer
84 views

Radius vs Radius of curvature of an ellipse

I am a bit confused by the physical meaning of radius vs radius of curvature, with regards to an ellipse. For a standard ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ In this case, the $a$ ...
0
votes
1answer
27 views

Is the observed widest-width of an oblate sphere constant under all rotations?

This is something which I feel intuitively is true but I'm having trouble finding a way of proving it mathematically. Given an oblate sphere, or ellipsoid, with equation $$x^2+y^2+(z^2 / c^2)=1, ...
2
votes
4answers
114 views

Area of triangle bounded by line and degenerate “crossed lines” conic

The question is Show that the two lines given by $$(A^2 - 3B^2)x^2 + 8ABxy +(B^2 - 3A^2)y^2=0$$ and the line given by $$Ax+By+C=0$$ determine an equilateral triangle of area ...
1
vote
1answer
50 views

Half parabola- $y = x^2$ - does the derivative exist at x = 0?

If we took $y = x^2$ and cut it in half by letting $x\ge 0$, does the derivative still exist at $x = 0$ or is it $\text{DNE}$? I think it's still $0$ because the function and it's derivative are both ...
0
votes
0answers
16 views

Test if a point is within a double intersected ellipse

I have a case of 4 ellipses, every 2 ellipses represent a pipe (outer and inner), and a front and back (back being occluded by the front) My question is that is there an easy way to obtain whether ...
0
votes
2answers
37 views

Circle $x^2+y^2=2$ is stretched by a scale factor $2$ parallel to the $x$-axis, find the equation of Ellipse

What is the quick method or formula to finding this answer? Also the method for finding the answer when the stretch is parallel to the $y$-axis, Regards Tom
0
votes
1answer
59 views

Rotating an ellipse about a line

I'm attempting to solve the following question: What is the volume of the region formed when the ellipse $9x^2+4y^2=36$ is revolved around the line $2x+y=5$? My try: $$9x^2+4y^2=36$$ $$y ...
1
vote
0answers
18 views

How to find foci from size and center?

How can I calculate the foci of an ellipse, given its width, height, and center? Everything I've found uses an equation instead of parameters.
0
votes
1answer
22 views

What is the formula/method used to show that $ b=4$ in this hyperbola?

Hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ Asymptotes $y=2x$ and $y=-2x$ Also given a point $A (2, 0)$ on the hyperbola (not sure if you need this though) I have absolutely no idea how you would ...
2
votes
2answers
99 views

Area of ellipse not in xy-plane

I've got a problem in which I'm trying to find the area of an ellipse which is given by the intersection of an elliptic cylinder with a plane. Nothing here is parallel to the coordinate axes, which is ...
0
votes
1answer
66 views

Finding equation of ellipse with given point and distance between directrices

I need to find the equation of an ellipse. The given were just a point where it passes, and distance between directrices. I know that the distance between directrices is given by $2a/e$. I don't ...
0
votes
1answer
38 views

A locus problem related to conic sections

Let $(C)$ be a circle of center $O$ and radius $r$. Let $E$ be any point on $(C)$. Let $P$ be any point other than $E$ in the plane. The perpendicular bisector of $[PE]$ cuts $(OE)$ in a point $M$. ...
1
vote
1answer
35 views

Is it true that these angles are equal?

Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to ...
0
votes
1answer
32 views

Find the equation of a hyberboloid with given base, narrowest section, and the distance between them

I have one question left in an assignment and I havn't been able to solve it. I know the equaton for a hyperboloid and I know that $a$ and $b$ will be equal to each other. I don't know how to solve ...
1
vote
1answer
206 views

Equation of a parabola in 3D space

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is vertex(lowest point) of the parabola. I only know z-coordinate of this point. I need to find coordinates ...
3
votes
2answers
108 views

How do I deal with reflections inside an ellipse?

Suppose I have an ellipse with foci $F_1$ and $F_2$. How do I show that any ray of light which intersects the segment connecting the foci will have subsequent reflections that always are tangent to ...
0
votes
0answers
17 views

Length of a right triangle's hypoteneuse projected onto a sphere

Please forgive me if this is the wrong kind of question, but I need someone to verify or refute my work. One leg of a triangle has length, $b$ (base), resulting from angle theta swept out by a ray ...
-3
votes
1answer
32 views

Calculating the circumference of an ellipse

I have searched for the answer and always find that there is an open solution (integral). C1 = INTEGRAL(0->2pi) sqrt(a^2 * cos^2[t]+b^2 *sin^2[t]) dt. (a nicer page is here: ...
1
vote
1answer
74 views

Intersecting two parabolas and computing the angle between the tangents in a point of intersection

I was solving some problems on parabola. I saw a question and solved it, but my solution was way too big. The question was: If $$\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} = ...
0
votes
1answer
56 views

Equation of normal vector pointing away from ellipse

Assuming that I have an ellipse, centered at $(h,k)$ of type: $$\left(\frac {x-h}{a}\right)^2 + \left(\frac {y-k}{b}\right)^2 = 1$$ The gradient of the normal is: $$\frac{a^2(y-k)}{b^2(x-h)}$$ ...
1
vote
0answers
63 views

Proof 5 points determine a conic without projective geometry

So I'm trying to prove that any five points, of which no 3 are colinear, there is a single conic that passes through al of them. I don't want to use projective geometry but rather, only analytic ...
0
votes
1answer
77 views

Finding volume of enclosed region

The base of S is the region enclosed by the parabola $y = 9 − 9x^2$ and the X - axis. Cross-sections perpendicular to the X - axis are isosceles triangles with ...
2
votes
1answer
48 views

Geometry of the Quadratic Formula

I am well aware of proofs of the quadratic formula that show, by completing the square and other methods, that the quadratic formula is what it is. I have been scouring the Internet and other ...
0
votes
0answers
30 views

Calculate parameters of an ellipse from the analytic form

The Wikipedia provides this equation: for an ellipse, defining with a, b - half axes, x_C, y_C - Position of the middle point, Theta - Rotation. Now I need to transform this other way ...
7
votes
2answers
80 views

'3-point' curve

If you have a loop of string, a fixed point and a pencil, and stretch the string as much as possible, you draw a circle. With 2 fixed points you draw an ellipse. What do you draw with 3 fixed points?
0
votes
2answers
73 views

What is the minimum information required to define an equation for ellipse?

What is the minimum information ie. amount of points in 2-dimensional plane in order to define the equation for an ellipse? I know that unique ellipse cannot be defined when only one of the foci is ...
1
vote
1answer
108 views

Find equation of ellipse given two tangent lines at given points and a point on ellipse

I'm attempting to generate an ellipse for a stair simulation game of mine, and the inputs are: A point on the ellipse The slope of the tangent line to the ellipse at that point Another point ...
2
votes
1answer
28 views

Problem with the type of equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ and vertices?

I am asked to find the type equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ , represents ? i.e a parabola , or hyperbola or ellipse or circle by squaring twice? Now , what I have done is like this ...
1
vote
2answers
53 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
1
vote
2answers
76 views

Is it possible to find equation for ellipse when focus, eccentricity and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...
0
votes
0answers
15 views

Point on ellipse after walking a distance on the perimeter [duplicate]

I've the equation of an ellipse. Given a point (x,y) on the ellipse and a length L , I want to find the coordinates (x1,y1) of the point where I'd end up after taking a walk of length L from (x,y), ...
5
votes
1answer
92 views

Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
0
votes
0answers
18 views

Calculating the position of the ascending and descending nodes in an orbit

I'm working on a space sim and I'm stuck getting the position of the Ascending and Descending orbital nodes. I know how to get the position at any angle of the orbit, but it's measured from the ...
0
votes
2answers
21 views

Need help with a conic tangent question? (Hyperbolics)

I need to find the equation of the tangent to the hyperbola $$\frac{x^2}{6}-\frac{y^2}{8}=1$$ at the point $(3,2)$. I tried doing it by substituting for $y$ but the algebra is not nice at all and I ...
1
vote
5answers
42 views

How to find the tangent to a conic?

I have this question. Find the equation of the tangent to the line $y^2=x$ at the point $(16,-4)$. I have tried to use both methods to work it out. 1) Substitute $y=mx+c$ into $y^2=x$ and find a ...
2
votes
0answers
32 views

Extremal points relative to origin for an ellipsoid

Suppose I have an ellipsoid of the form $ax^2 + by^2 + az^2 - cxy -cyz = d$ How would I find the points nearest to, and furthest from, the origin?
0
votes
1answer
50 views

Problem involving conics. Need to find points of intersection given information about a conic.

A conic has eccentricity $e=0.7$, a focus $(5,−3)$ and directrix $y=2x−7$. Find the points of intersection of the conic with line $y=−3$. I'm really stuck on this, and have no idea even where to ...
0
votes
1answer
62 views

Help with Conic: Hyperbola's chord of contact

please help with this proof. "Show that the tangents at the endpoints of a focal chord of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ meet on the corresponding directrix." This is a ...
1
vote
0answers
38 views

What is the intersection between $x + y - z = -2$ and $z^2 = x^2 + y^2$

I got the answer as $4x + 4y + 2xy + 4 = 0$ by substituting $z = x + y + 2$ into the second equation, but I feel as this is wrong since I am missing $z$ in the function. How do I approach this ...
0
votes
3answers
189 views

Detect if two ellipses intersect

I have seen a lot of papers on how to find points of intersection between two ellipses for 2D case, but i only need to check if two ellipses are in collision. I don't need to know points of ...