Conic Sections (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · Gaokao

Conic Sections: core idea

Intersection of conics.

The locus $|z-1|=2|z+1|$ is a circle in the complex plane. What is the $x$ -coordinate of its center?

Hint: Square both distances and complete the square. Answer: -5/3.

1. How many lines are tangent to both concentric circles $x^2+y^2=1$ and $x^2+y^2=4$ ?
2. Polar of $(2,0)$ w.r.t. $x^2+y^2=1$ : $x=$ ?
3. Parametric parabola $y^2=4ax$ , point $t=1$ , $a=1$ : $x$ ?
4. Find the eccentricity of the hyperbola $x^2-y^2=a^2$ .
5. Find the radius of the circle $x^2+y^2=2x$ .
6. Area of ellipse $\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$ ?
7. Shortest distance from $(0,0)$ to $x^2-y^2=1$ ?
8. Director circle of $\dfrac{x^2}{25}+\dfrac{y^2}{9}=1$ : radius?
9. Pair of tangents from $(5,0)$ to $x^2+y^2=9$ : length of tangent?