Problem Bank

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has left vertex $A(-2,0)$ and eccentricity $e=\dfrac{\sqrt 3}2$.

1. Find the standard equation of the ellipse $C$. 2.…

Consider the hyperbola $E \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ and a point $M(x_0, y_0)$ not on $E$ with $x_0y_0 \neq 0$. Let $N(\lambda x_0, \lambda y_0)$, where

A sequence $\{a_n\}$ satisfies $a_1=\dfrac 12$ and

1. Prove that $1\leqslant\dfrac{a_n}{a_{n+1}}\leqslant 2$ for all $n\in\mathbb N…

The circle $x^2 + y^2 - 6y + m = 0$ and the line $x + 2y - 3 = 0$ meet at points $P$ and $Q$, and the circle with diameter $PQ$ passes through the origin. Find $m$.

Let $f(x) = (x-1)e^x - ax + 1$.

1. For $a = 0$, solve the inequality $f(x) \leqslant 0$. 2. Prove that for every $a > 0$,

3. Restrict $f$ to th…

Let $f(x) = \mathrm{e}^{x-a} - ax$ with $a \in \mathbb{R}$.

1. If $f(x)$ has two zeros, find the range of possible values of $a$. 2. If for every $x \in [0, +\infty)$,

In $\triangle ABC$, the perpendicular bisector of side $BC$ meets $BC$ at $D$ and $AC$ at $M$. If $\overrightarrow{AM}\cdot\overrightarrow{BC} = 6$ and $AB = 2$, find $AC$.

A sequence $\{a_n\}$ has $a_1 = 2$ and

where $S_n$ is the partial sum. Find the maximum value of $S_n$.

Solve over the complex numbers:

For what value of $t$ is the ellipse

tangent to the hyperbola $xy = 1$?

In the coordinate plane $xOy$, let $A,B,C$ be distinct points on the circle $x^2+y^2=1$. If there exist reals $\lambda,\mu$ with $\overrightarrow{OC}=\lambda\overrightarrow{OA}+\mu…

Let $0 < a < b < \dfrac{1}{\mathrm{e}}$. Arrange

in increasing order.