Conic Sections Problems: Get Step-by-Step Solutions

Name or Adjective for Ellipse with very different (or very similar) scales
I am looking for an adjective or word to describe an ellipse (or ellipsoid, in more dims) where the length of the principal axes are of roughly the same order of magnitude $O (a) = O (b)$ (including equal, i.e. a circle).
For instance:
- $a = 2, b = 5$ would be something like an "isometric" ellipse (or ellipsoid)
- $a = 2, b = 50$ would be something like an "asometric" ellipse (or ellipsoid) since they are of different orders of magnitude

Halle Marsh 2022-04-22

If x and y are real numbers and $4 x^{2} + 2 x y + 9 y^{2} = 100$ then what are all possible values of $x^{2} + 2 x y + 3 y^{2}$ .

zergingk8l 2022-04-21

How to find the position on ellipse (or hyperbola) arc if we know it's euclidean distance from given point and direction of movement?

Davin Sheppard 2022-04-21

How to find the axes of a rotated ellipse
I have a set of 17 points which I know are on an ellipse. I have the x,y co-ordinates of each point; the y-axis is vertical and the x-axis is horizontal. I want to prove these points are on an ellipse, but the ellipse is rotated clockwise by approximately 14 degrees (determined visually - I want to calculate the exact value of the rotation). I need to find the exact position of the major and minor axes (x' and y') and I do not know the values of the semi-major axis (a) or the semi-minor axis (b). Is this possible?
I have tried to find the x,y co-ordinates of the points furthest from (and nearest to) the centre of the ellipse, but I've only managed this through very many tedious iterations and it isn't exact -- is there a better way? Thank you.

Jakobe Norton 2022-04-19

What is the complex form through five points of a conic section in the complex plane?

Ormezzani6cuu 2022-04-18

What can you say about the motion of an object with velocity vector perpendicular to position vector? Can you say anything about it at all?
I know that velocity is always perpendicular to the position vector for circular motion and at the endpoints of elliptical motion. Is there a general statement that can be made about the object's motion when the velocity is perpendicular to position?

Kale Bright 2022-04-18

What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic sections form?
How will look the general conic sections equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ in case of rotated and shifted from coordinates origin ellipse, parabola and hyperbola?
I need a formulas for coefficients A, B, C, D, E and F for ellipse, hyperbola and parabola. I did it for not-rotated conic sections at the origin of coordinates but have a difficults with rotated and shifted.
For i.e. standart ellipse equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ gives me general equation $\frac{1}{a^{2}} x^{2} + 0 x y + \frac{1}{b^{2}} y^{2} + 0 x + 0 y - 1 = 0$
I need the same in case if ellipse located in position (h;k) and rotated on some angle $α$ from positive X axis. And the same for parabola and hyperbola.

Henry Winters 2022-04-16

Is $x^{2} = 4 a y$ a function while $y^{2} = 4 a x$ is not?

Willow Cooper 2022-04-14

Find semi-major/semi-minor axis of ellipse from parametric equations with different phase
$x = \hat{x} \cdot \cos (Ω t - θ)$
$y = \hat{y} \cdot \sin (Ω t - φ)$

Colten Welch 2022-04-13

Is there a parametrization of a hyperbola $x^{2} - y^{2} = 1$ by functions x(t) and y(t) both birational?
Consider the hyperbola $x^{2} - y^{2} = 1$ . I am aware of some parametrizations like:
1. $(x (t), y (t)) = (\frac{t^{2} + 1}{2 t}, \frac{t^{2} - 1}{2 t})$
2. $(x (t), y (t)) = (\frac{t^{2} + 1}{t^{2} - 1}, \frac{2 t}{t^{2} - 1})$
3. $(x (t), y (t)) = (cosh t, sinh t)$
4. $(x (t), y (t)) = (\sec (t), \tan (t))$
The first and the second are by rational functions x(t) and y(t). But the functions are not birational(i.e. with rational inverse of each).
Is there a parametrization where:
- x(t) is rational with inverse also rational, and
- y(t) is rational with inverse also rational?
Is possible, to find a parametrization where both are rational and at least one of the has inverse rational?

Achatesopo8 2022-04-12

Find the parametric equation for the tangent line to the intersection curve between an ellipsoid and a paraboloid?
We got various problems in this site asking for similar problems btw an ellipsoid and a plane. What if it's btw an ellipsoid and a paraboloid?
I got the equation of both surfaces: $4 x^{2} + y^{2} + z^{2} = 9$ and $z = x^{2} + y^{2}$ respectively. Now given the tangent line touches their intersection (as an off-origin xz-ellipse but this might be useless) at (-1,1,2), how can we find the v- for the parametric equation of the tangent line $\underset{―}{r} (t) = (- 1, 1, 2) + t \underset{―}{v}$ ? Is there any way we need to utilize the directional derivative of either surface? Or we just use f(x,y,z) and g(x,y,z) to describe both surfaces so that $\underset{―}{v} = \nabla f \times \nabla g$ which I got after simplification as (5,8,6)? Thanks in advance.

Jewel Beard 2022-04-12

Find domain and range of the slanted hyperbola
Given the conic section $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ and I know that it is a hyperbola and $B \neq 0$ .
How to find its domain and range? I guess the method of Lagrange multipliers will fail here.

Destinee Bryan 2022-04-10

What is the easier way to find the circle given three points?
Given three points $(x_{1}, y_{1}), (x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ , if
$\frac{y_{2} - y_{1}}{x_{2} - x_{1}} \neq \frac{y_{3} - y_{2}}{x_{3} - x_{2}} \neq \frac{y_{1} - y_{3}}{x_{1} - x_{3}},$
then there will be a circle passing through them. The general form of the circle is
$x^{2} + y^{2} + d x + e y + f = 0 .$
By substituting $x = x_{i} and y = y_{i}$ , there will be a system of equation in three variables, that is:
$\begin{aligned} (\begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix}) (\begin{matrix} d \\ e \\ f \end{matrix}) & = (\begin{matrix} - (x_{1}^{2} + y_{1}^{2}) \\ - (x_{2}^{2} + y_{2}^{2}) \\ - (x_{3}^{2} + y_{3}^{2}) \end{matrix}) . \end{aligned}$
As there are a lot of things going around, the solution is prone to errors. Maybe this solution also has an error.
Is there a better way to solve for the equation of the circle?

painter555ui8n 2022-04-07

Fitting a ballistic trajectory to noisy data where both spacial and temporal domains observations are noisy
Fitting a curve to noisy data is somewhat trivial. However it generally assumes that data abscissa is fixed, and the error is computed on the ordinate.
In my setup, I have 3D spacial observations of ballistic trajectories (that I model with a simple parabola), but the observations time are also noisy.
Therefore, I have to estimate the initial position $y_{0}, y_{0}, z_{0}$ and initial speed $v_{x_{0}}, v_{y_{0}}, v_{z_{0}}$ , based on 4D (noisy) observations $(X_{i}, Y_{i}, Z_{i}, T_{i}), i \in [0, N]$ , such that they fit the following model:
${\begin{aligned} x (t) & = x_{0} + v_{x_{0}} t \\ y (t) & = y_{0} + v_{y_{0}} t \\ z (t) & = z_{0} + v_{z_{0}} t - \frac{g}{2} t^{2} \end{aligned}$
with t monotonically increasing with i.
I'm not sure how to formulate such optimization problem because I have 6 parameters to estimate, but also 4N variables with only 3N equations… My intuition tells me there's only one single parabola that minimizes the error (MSE for example), but I can't formulate the problem.

Jaylyn Villarreal 2022-04-06

Finding equation of a path in the plane $y = z$
What is the easiest way to see that the path $\underset{―}{r} : R \to R^{3} : t \mapsto (\sin, t, \cos, t, \cos, t)$
traces out an ellipse in the plane $y = z$ ?
I think first rotating $R^{3}$ by $\frac{π}{4}$ about the x-axis will help but I am not sure how to proceed.

zdebe5l8 2022-04-05

Finding the axis and orientation of an ellipse with matrices
So I have this ellipse equation:
$5 x^{2} + 10 y^{2} - 12 x y = 14$
I'm asked to get the lengh of the semi-major and semi-minor axis, and it's orientation.

mwombenizhjb 2022-04-04

Equation of a section plane in hyperbolic paraboloid
Find the equation of a plane passing through Ox and intersecting a hyperbolic paraboloid $\frac{x^{2}}{p} - \frac{y^{2}}{q} = 2 z (p > 0, q > 0)$ along a hyperbola with equal semi-axes.
My attempt: The equation of a plane passing through Ox is $B y + C z = 0$ . So $z = \frac{- B y}{C}$ . Now substitute $z = \frac{- B y}{C}$ into the equation of hyperbolic paraboloid $\frac{x^{2}}{p} - \frac{y^{2}}{q} = \frac{- 2 B y}{C}$ and transform to $\frac{x^{2}}{p} - \frac{{(y - \frac{B q}{C})}^{2}}{q} = - \frac{B^{2} q}{C^{2}}$ . What to do next? I don't understand how to find B and C if we assume $\frac{x^{2}}{p} - \frac{{(y - \frac{B q}{C})}^{2}}{q} = - \frac{B^{2} q}{C^{2}}$ is a hyperbola with equal semi-axes.

Nathen Peterson 2022-04-03

Eliminate the parameter t from
$h = \frac{3 t^{2} - 4 t + 1}{t^{2} + 1}$
$k = \frac{4 - 2 t}{t^{2} + 1}$

ab0utfallingm1z2 2022-04-01

I have the general equation
$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$
and I want to determine what conic section it represents according to the value of its coefficients or relations between them, is there a theorem where that is explained?

When you are told to work with conic sections equations, think about how planets travel around the sun and how elliptical routes get in focus. Parabolic mirrors, solar explanations, and so on. We have conic sections practice problems with answers that will help you learn easier. In the majority of cases, one should start with conic section equations because it is where one must start. Alternatively, take your conic section equation and compare things to the answers that we have below. See provided conic sections examples as these solutions will provide you with great starting points.

Integral Calculus

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