Found problems: 137
In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic.
Proposed by [i]Alex Wang[/i]
Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.
Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.
Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt2$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.
Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^{2}$ with boundary $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A)\geq \frac{\pi}{2}$.
The tangents at four different points of an arc of a circle less than $180^o$ intersect forming a convex quadrilateral $ABCD$. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.
Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$
(1) Find the condition for which $C_1$ is inscribed in $C_2$.
(2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$.
(3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$.
60 point
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(a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value.
(b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere.
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[i]Proposed by Tran Quoc Binh[/i]
Let \(\mathcal E\) be an ellipse with foci \(F_1\) and \(F_2\), and let \(P\) be a point on \(\mathcal E\). Suppose lines \(PF_1\) and \(PF_2\) intersect \(\mathcal E\) again at distinct points \(A\) and \(B\), and the tangents to \(\mathcal E\) at \(A\) and \(B\) intersect at point \(Q\). Show that the midpoint of \(\overline{PQ}\) lies on the circumcircle of \(\triangle PF_1F_2\).
[i]Proposed by Karthik Vedula[/i]
Prove that the simultaneous equations
$$x^4 -x^2 =y^4 -y^2 =z^4 -z^2$$
are satisfied by the points of $4$ straight lines and $6$ ellipses, and by no other points.
A segment $AB$ is given. Let $C$ be an arbitrary point of the perpendicular bisector to $AB$; $O$ be the point on the circumcircle of $ABC$ opposite to $C$; and an ellipse centred at $O$ touch $AB, BC, CA$. Find the locus of touching points of the ellipse with the line $BC$.
$F_1,F_2$ are two focal points of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, $P$ is a point on the ellipse, and $|PF_1|:|PF_2|=2:1$, then the area of $\triangle PF_1F_2$ is________.