Q.Find the equation of the set of all points whose distance from $(0, 4)$ are $\dfrac{2}{3}$ of their distance from the line $y = 9$.

Ellipse

The standard ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ (with $a>b$) has centre at

the origin, major axis $2a$, minor axis $2b$, and eccentricity $e$ given by

$b^2=a^2(1-e^2)$. Its foci are $(\pm ae,0)$, directrices $x=\pm\dfrac{a}{e}$, and the

latus rectum has length $\dfrac{2b^2}{a}$. Every point satisfies the focal-distance

property $SP+S'P=2a$.

The line $y=mx+c$ is a tangent iff $c^2=a^2m^2+b^2$, so tangents of a given slope

are $y=mx\pm\sqrt{a^2m^2+b^2}$; the tangent at $(a\cos\theta,b\sin\theta)$ is

$\dfrac{x\cos\theta}{a}+\dfrac{y\sin\theta}{b}=1$. The position of a point

$(x_1,y_1)$ is decided by the sign of $S_1=\dfrac{x_1^2}{a^2}+\dfrac{y_1^2}{b^2}-1$

(inside if $<0$). Shifting the centre to $(h,k)$ replaces $x,y$ by $x-h,y-k$. These …