Articles of conic sections

Equation of family of parabolas having fixed directrix.

The question is actually to find the order of the differential equation of this family, but I cannot formulate the equation. I searched a little and found the equation to be something like $(y-k)^2=4ax$. But if we assume the directrix to be the line x = -a and take a specific case when k=0, then […]

Parabola and Circle problem : The parabola $y =x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn …

Problem : The parabola $y=x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. Find the length at a tangent to the circle from O. My approach : Since the parabola $y=x^2-8x+15$ cuts the x axis therefore, its y coordinate is […]

Determining the normal of an ellipse

Given I have (in a 2D coordinate system) an ellipse with the center at $(c_x,c_y) = (0,0)$ where I do not know the actual value of the major an minor axis but I have the ratio $r=\frac{a}{b}$ and an inclination angle $\phi$. If I know that the point $p = (p_x,p_y)$ are on the ellipse […]

Equation of diagonal ellipse knowing 2 foci and eccentricity

I am trying to find the equation of a diagonal ellipse knowing the position of the two focus points and the eccentricity. Online I can only find the equation of the ellipse where the two foci are located on the same y axis value. Any idea on how to do this? Thank you.

Find the circle circumscribing a triangle related to a parabola

This question already has an answer here: A question on the parabola.. 1 answer

Conditions for intersection of parabolas?

What are the conditions for the existence of real solutions for the following equations: $$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$ where $a,b,c,d $ are real numbers. These represent two parabolas; how might we find out the conditions for the existence of $0,2,4$ real solutions of the equations?

Locus of intersection of two perpendicular normals to an ellipse

I came up with the following question while playing around with Geogebra, and rather than do it myself I figured I’d offer it here. For a given ellipse, the director circle is defined to be the set of all points where two perpendicular tangents to the ellipse cross each other. Suppose we instead consider the […]

general equation of a tangent line to a hyperbola

Suppose that there is a hyperbola of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$. I would like to figure out an equation that describes tangent line to this hyperbola. How would I be able to do this using calculus? My calculus trials are bring me some gibberish answers. I did use the calculus below, but what I […]

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described by $z$ is? I attempted to rewrite this in cartesian form but to no avail. How do i proceed?

Minimal number of points to define a rotated ellipse?

What is the minimal number of points $N$ to uniquely define the semi-major axis $a$, the semi-minor axis $b$ and the rotation angle $\omega$ of an ellipse whose the center is known/fixed (this is related to ellipse fitting procedures). In other words, if we consider that the center is known/fixed, on this image what is […]