Problem Bank

The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has eccentricity $\dfrac{\sqrt2}{2}$; two non-vertex points $M, N$ on it satisfy $\angle MON = 90^\circ$ and

Consider the ellipse $C \colon \dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$. The line $l \colon my = x + 3m - 2$ passes through a fixed point $P$ and meets $C$ at two points $A, B$ with $|…

In the plane of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$), let $P(x_0,y_0)$ be a point other than the origin. Through $P$, draw two arbitrary secant lines $AB$ …

Compute

A gym is $2$ km from home. Starting from home, a walker covers $\tfrac34$ of the way to the gym, then changes his mind and walks $\tfrac34$ of the way back home, then $\tfrac34$ of…

Two right triangles $ABC$ and $ADC$ share the common hypotenuse $AC = 10$ and lie on opposite sides of it, so that $\…

Sequences $\{x_n\}, \{y_n\}$ satisfy $x_0 = 5$, $y_0 = 2$, and

Find $\lim_{n\to\infty} x_n$ and $\lim_{n\to\i…

The hyperbola $C \colon \dfrac{x^2}{4} - y^2 = 1$ has left and right vertices $A_1, A_2$. A line $l$ through $P(4, 0)$ meets the right branch of $C$ at two points $M$ and $N$.

1. …

Let $f(x) = \dfrac{1}{x}$ and $g(x) = ax^2 + bx$, where $a, b \in \mathbb{R}$ and $a \neq 0$. Suppose the graphs of $y = f(x)$ and $y = g(x)$ have exactly two common points $A(x_1,…

Let $a \in \mathbb{R}$ and

1. When $a = 1$, prove that $f(x) > \mathrm{e}$. 2. If $f(x) > \m…

The ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has left and right foci $F_1$, $F_2$, eccentricity $\dfrac{\sqrt 2}2$, and minor axis of length $2$. Let $A,B,C,D$ be …

Let $C$ be the locus of points in the plane whose distances to the fixed point $F(0, -2)$ and to the fixed line $l \colon y = 2$ sum to $6$. For a point $P$ on $C$ and a given poin…