Free solutions to conic sections equations

$(2 x + \sqrt{4 x^{2} + 1}) (\sqrt{y^{2} + 4} - 2) \geq y > 0$ minimum vale of $x + y$

abiejose55d 2022-05-01

Shape induced by the condition of angle bisector passing through a fixed point
Let A, B be two points on 2D plane. For any $C \in \overset{―}{A B}$ , define the set $S = {P; ∣; ∠ A P C = ∠ C P B} .$
What is the shape of S?

hadaasyj 2022-05-01

Rotation matrix to construct canonical form of a conic
$C : 9 x^{2} + 4 x y + 6 y^{2} - 10 = 0 .$
I've found C is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've studied the characteristic polynomial
$p (t) = \det (\begin{matrix} 9 - t & 2 \\ 2 & 6 - t \end{matrix})$
The eigenvalue are $t_{1} = 5, t_{2} = 10$ , with associated eigenvectors $(- 1, 2), (2, 1)$ . Thus I construct the rotation matrix R by putting in columns the normalized eigenvectors (taking care that $det (R) = 1$ ):
$R = \frac{1}{\sqrt{5}} (\begin{matrix} 1 & 2 \\ - 2 & 1 \end{matrix})$
Then ${(x, y)}^{t} = R {(x^{'}, y^{'})}^{t}$ , and after some computations I find the canonical form
$\frac{1}{2} x^{' 2} + \frac{4}{5} y^{' 2} = 1 .$

NepanitaNesg3a 2022-04-30

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?

2022-04-27

znacimavjo 2022-04-25

How can you prove that the midpoint of the square will be Origin?
Vertices A,B,C,D and the midpoints of the sides of the square ABCD lie on $x^{2} y^{2} = 1$ .
No other points of the square lies on the curve.
How can you prove that the midpoint of the square will be Origin ?
My approach: If I can show if (x,y) are the coordinates of A then (y, -x) will be so of B , I will be done. I supposed A lies on first quadrant , B lies on second quadrant etc..
Can anyone tell please me how to show it ?

Lymnmeatlypamgfm 2022-04-24

How to solve system of equations $A_{1} x^{2} + B_{1} x y + C_{1} y^{2} + D_{1} x + E_{1} y + F_{1} = 0$ and $A_{2} x^{2} + B_{2} x y + C_{2} y^{2} + D_{2} x + E_{2} y + F_{2} = 0$
How to express x through y?

bacfrancaiso0j 2022-04-23

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

Maeve Bowers 2022-04-23

Given: $y = x^{1.65}$
Could it be described as parabolic in shape, or does the equation have to have $x^{2}$ as its highest degree term?

hapantad2j 2022-04-23

Geometric meaning of $| | z - z_{1} | - | z - z_{2} ∣ ∣ = a$ , where $z, z_{1}, z_{2} \in C$

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 $x^{2} + 2 x y + 3 y^{2}$ .

Naomi Hopkins 2022-04-22

Formula for analytical finding ellipse and circle intersection points if exist
I need a formula that will give me all points of random ellipse and circle intersection (ok, not fully random, the center of circle is laying on ellipse curve)
I need step by step solution (algorithm how to find it) if this is possible.

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?

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.

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Conic Sections Questions & Answers