Conic Sections Problems: Get Step-by-Step Solutions

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
$\frac{1}{2} x^{' 2} + \frac{4}{5} y^{' 2} = 1 .$

Hugh Soto 2022-03-31

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?

Polchinio3es 2022-03-30

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

Oxinailelpels3t14 2022-03-29

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?

Pizzadililehz 2022-03-27

Coordinate geometry and Trignometry.
Find the condition so that the line $p x + q y = r$ intersects the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in points whose eccentric angles differ by $\frac{π}{4}$ .
Though I know how to solve it using parametric coordinates, I was wondering if there's an another approach which is less time consuming.

Coradossi7xod 2022-03-27

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?

ropowiec2gkc 2022-03-26

Determine whether $(x, y) = (m \cos (ω t), n \cos (ω t - ϕ))$ describes an ellipse for all m, n, $ω, ϕ$

rijstmeel7d4t 2022-03-26

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 ?

zatajuxoqj 2022-03-24

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.

Oxinailelpels3t14 2022-03-24

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

I want to know how to find the vertices of the conic equation
20x^2 +4y^2-800=0

Jeffrey Ali 2022-03-18

A square ABCD has all it's vertices on $x^{2} y^{2} = 1$ . The midpoints of it's sides also lie on the same curve . What is the area of this square?
It can be found very easily if we can show that diagonal of this square meet at the origin.

Abbie Edwards 2022-03-17

Can anyone help with paramaterization of conics?
Im struggling to wrap my head around an example. It considers the conic $x^{2} + y^{2} - z^{2} = 0$ then proceeds:
Take $A = [1, 0, 1]$ and the line P(U) defined by $x = 0$ . Note that this conic and the point and line are defined over any field since the coefficients are 0 or 1. A point $X \in P (U)$ is of the form $X = [0, 1, t]$ or [0, 0, 1] and the map $α$ is
$\begin{aligned} α ([0, 1, t]) & = [B ((0, 1, t), (0, 1, t)) (1, 0, 1) - 2 B ((1, 0, 1), (0, 1, t)) (0, 1, t)] \\ = [1 - t^{2}, 2 t, 1 + t^{2}] \\ or α ([0, 0, 1]) = [- 1, 0, 1] . \end{aligned}$

How do I evaluate B(v,v) or B(v,v)(a,b,c) like they have to go from the first line to the second?

maxime99bl6 2022-03-15

Angle between normal vector of ellipse and the major-axis.
I am trying to derive the angle made between the major or x-axis and the normal vector of an ellipse of general shape $x = a \cos (t), y = b \sin (t)$ with the parameter t reffering to Ellipse in polar coordinates. I need to solve it for any angle t. From standard reasoning I find the normal vector by its definition and checked it with the page on mathworld from wolfram and works well. Then since I know 2 points, namely a point ON the shape and a point on the normal vector I derive the angle of interest to be $\tan (ϕ) = \frac{a}{b} \tan (t)$ Derivation. However this is very similar to the polar angle namely its simply the term a and b flipped. But when thinking about it I keep getting confused, am I correct or do I need the polar angle? If so where did I go wrong?
I also found Normal to Ellipse and Angle at Major Axis but this page confused me a bit, one idea I had was they use the polar angle vs the angle I am in need of $(ϕ)$ then I would indeed get by combing $t = \tan^{- 1} (\frac{a}{b} \tan (θ))$ and $ϕ = \tan^{- 1} (\frac{a}{b} \tan (t))$
$ϕ = \tan^{- 1} (\frac{a}{b} \tan (\tan^{- 1} b i g (\frac{a}{b} \tan θ b i g))) = \tan^{- 1} (\frac{a^{2}}{b^{2}} \tan θ)$
My excuse for my rambling, I find these angles confusing...

Jeffrey Ali 2022-03-15

Algorithm to determine if a 3D ellipsoid is contained within another?
Can anyone point me to an algorithm for how to efficiently check if a 3D ellipsoid is contained within another one? We can assume their origins are collocated.
I am dealing with covariance ellipsoids constructed from matrices.

brecruicswbp 2022-03-09

Why is this solution incorrect?
Prove that on the axis of any parabola there is a certain point K which has the property that, if a chord PQ of the parabola be drawn through it, then $\frac{1}{P K^{2}} + \frac{1}{Q K^{2}}$ is the same for all positions of the chord.
If it would be valid for standard parabola than it would be valid for all parabolas. Thus, proving for $y^{2} = 4 a x$ .
let the point K be (c,0)
Equation of line PQ using parametric coordinates: $x = c + r \cos θ$ (Equation 1), $y = r \sin θ$ (Equation 2)
From equation 1 and 2: ${(x - c)}^{2} + y^{2} = r^{2}$ (Equation 3)
Using Equation 3 and $y^{2} = 4 a x$ , we get this quadratic in r: $r^{2} - 4 a x - {(x - k)}^{2} = 0$ (Equation 4)
Roots of this quadratic $(r_{1} and r_{2})$ would be the lengths of PK and QK
From Equation 4, $r_{1} + r_{2} = 0$ and $r_{1} r_{2} = - (4 a x + {(x - k)}^{2})$
We know, $\frac{1}{P K^{2}} + \frac{1}{Q K^{2}} = \frac{1}{r_{1}^{2}} + \frac{1}{r_{2}^{2}} = \frac{{(r_{1} + r_{2})}^{2} - 2 r_{1} r_{2}}{{(r_{1} r_{2})}^{2}} = \frac{- 2}{r_{1} r_{2}}$
As value of $r_{1} r_{2}$ is not constant thus $\frac{1}{P K^{2}} + \frac{1}{Q K^{2}}$ does not turn out to be constant. Hence, this solution is incorrect.
I've seen the correct solution but I wanted to know why this solution is incorrect?

Dax Gross 2022-03-09

Why tangents to a quadratic curve never cut it again?
I have been studying conic sections lately and couldnt figure out why we can use the condition $D = 0$ for the quadratic equation having only one root in many of the questions involving finding tangents, which I have read in certain places is called the envelope method. That got me to thinking why the tangent to the conic passes through one and only one point on the conic. A tangent is defined only as a line which just touches the curve at one point. However, it lays no restrictions on whether it could intersect the curve again or not.
So why exactly does a tangent to a circle/ ellipse/ parabola/ hyperbola never cut it again?

Kai Ashley 2022-02-12

How do you find the points of intersection of the curves with polar equations $r = 6 \cos θ and r = 2 + 2 \cos θ$ ?

Pegov1 2022-02-12

How do you find the eccentricity, directrix, focus and classify the conic section $r = \frac{14.4}{2 - 4.8 \cos θ}$ ?

kaibaloveyou3cj 2022-02-12

How do you find the eccentricity, directrix, focus and classify the conic section $r = \frac{0.8}{1 - 0.8 \sin θ}$ ?

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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