# Questions tagged [conic-sections]

For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.

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### Proof of why conics map to conics after a perspective transformation

Background Consider a world where the ground is the standard $x$-$y$ plane with a Cartesian grid on it. The graph of a parabola $x^2 = 4ay$ is on this $x$-$y$ plane. A person of with eye-level $h$ ...
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### Geometric proof for why the midpoints of parallel chords of a parabola lie on the same line parallel to the axis

I was trying to figure out a geometric proof for why the midpoints of parallel chords of a parabola lie on the same line which is parallel to its axis. I searched on StackExchange and people have ...
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### Packing parabola with the chain of internally tangent circles

Related to the question Condition for perfect packing of ellipse with circles along the major axis An attempt of "perfectly pack" a parabola $x=ay^2$ as a special king of ellipse with ...
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### Derivative of A Quadratic Form with a Change of Basis

The derivative of a quadratic form is given by: $\Delta_{X} X^{T}AX = 2AX$ where A is symmetric. What is the derivative of the following: $\Delta_{X} (DX)^{T}A(DX)$, where $A$ is positive and ...
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### Finding position of endpoint on an ellipse's perimeter given axes

Given the semiaxes and $d$ (where $d$ starts at the semimajor axis and ends at point $A$), is there a formula to calculate the position of the endpoint on the ellipse perimeter? In my case, the ...
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### Angular coefficient of the straight line tangent to the parabola at one of its points: alternative proof

Let be $P_0(x_0,y_0)\in\mathcal P : \, y=ax^2+bx+c$. We know that the $m$ angular coefficient of the straight line tangent $t$ in $P_0$ is: $$m=2ax_0+b \tag 1$$ To find the $(1)$ I before consider the ...
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### How to prove distance from foci on an ellipse is equal to twice the semi-major axis (for specific ellipse)

Prove that for any point (x,y) on the conic, the sum of the distances to the two foci is always twice the semi-major axis. I know that this can be proven in general for all ellipses but the practice ...
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### Tangent to an ellipse

I want to find the tangent on point $(x_0, y_0)$ to an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We assume $y_0$ is positive. We can derive $y=\frac b a \sqrt{a^2-x^2}$. In order to ...
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### Homographic function: alternative proofs to obtain $ad-bc$

Considering the function, $$y=\frac{ax+b}{cx+d}\tag1$$ If $c = 0 \wedge d\neq 0$, the function represents a straight line of equation $$y=\frac ad x+ \frac bd$$ If $c ≠ 0$ and $ad = bc$ the function ...
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### Given ellipse of axes $a$ and $b$, find axes of tangential and concentric ellipse at angle $t$

Let’s say I have an ellipse with horizontal axis $a$ and vertical axis $b$, centered at $(0,0)$. I want to compute $a’$ and $b’$ of a smaller ellipse centered at $(0,0)$, with the axes rotated by some ...
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### Find the eccentricity of the conic $4x^2+y^2+ax+by+c=0$, if it tangent to the $x$ axis at the origin and passes through $(-1,2)$

Solving this would require three equations (1) Tangent to x axis at origin Substituting zeroes in all $x$ and $y$ gives $c=0$ (2) Passes through (-1,2) $$4(1)+4-a+2b=0$$ $$-a+2b=-8$$ How do I find the ...
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### geometric derivation of formulas for conic sections from a double cone

I'm trying to find derivations for the ellipse and the hyperbola that use geometric proofs such as the one found for the parabola on wikipedia (image for clarity) I'm aware that there are proofs ...
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### How do I find the point on a given hyperbola that passes through a given point not on the hyperbola?

There's a family of lines and where they intersect, the envelope, is a hyperbola. The hyperbola's center is at the origin and the $y$-axis is its axis of symmetry, and I know both $a$ and $b$. I'm ...
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### The mirror point of an ellipse's focus about the tangent line through a point is collinear to said point and the other focus

I was reading a bit about ellipses, specifically why light sent out from one focus will reflect off the ellipse and converge back to the other focus. The proof I was reading involves a construction as ...
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### How to find the point of intersection of normals to a general conic drawn from points where tangents from an external point P meet the conic?

Let's say we are given a general conic's eqn: $ax^2+by^2+2gx+2fy+2hxy+c=0$, and also a fixed point $P(r,s)$. Now I draw a pair of tangents to the conic from $P$. Let these tangents meet the conic at ...
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### 3 new Points lying on Jerabek Hyperbola?

R is the circumcenter, H is the orthocenter of the triangle ABC. Then points F,G,E are the points of intersection of the altitudes with the sides of the triangle ABC. U, V, W are the intersection ...
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### Triangle tangent to 3 Parabolas, finding the common area

Triangle $ABC$, $AB=4$, $BC=15$, $AC=13$. Two sides are tangents to the respective Parabolas. We have to find the area shaded. My approach- I tried finding the area of the quadratures(Archimedes) ...
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### Area between parabola and a line that don't intersect? 0 or infinity

Came across a problem on social media, Find the area of the region bounded by a parabola, $y = x^2 + 6$ and line a line $y = 2x + 1$. I tried to draw it on paper and they didn't seem to intersect. ...
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### Ellipse on complex plane

I have found that the curve $z(x)$ on the complex plane with $$z=\frac{ax+b}{x^2+1}$$ at real $x$ spanning from $-\infty$ to $+\infty$ looks very similar to an ellipse at any complex parameters $a,b$...
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### The locus of vertex of a parabola, given that an orthogonal intersection is made with another having specified latus rectum and orientation.

Full Question: A variable parabola of fixed latus rectum 4b and having axis parallel to x–axis, lies completely in Ist and IVth quadrant and cuts the fixed parabola $y^2=4ax$ orthogonally. The locus ...
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### Finding the intersection of two cones (modeling an inflatable structure formed from truncated cones)

I am trying to model the join of the leading edge of this 'wing'. The tubes are inflated, each section is conical in shape. I know the radius of the tubes at each joint. Given a vector in 3D space of ...
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### Orthocenter of triangle with vertices on the hyperbola is also on the hyperbola

The points $P(p, 1/p), \,Q(q,1/q), \,R(r, 1/r)$ and $S(s, 1/s)$ lie on the curve $xy = 1$. a) If $PQ || RS$, show that $pq = rs$. b) Show that $PQ\perp RS$ if and only if $pqrs=-1$. c) Use part b to ...
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### How to find the required equation for a family of curves?

The question given is: Tangents are drawn from two points $(x_1,y_1)$ and $(x_2,y_2)$ to $xy=c^2$. The conic passing through the two points and through the four points of contact will be a circle, ...