Question

Find the equation to the locus of a point which moves so that the sum of its distance from $(3,0)$ and $(-3,0)$ is less than $9$.

Answer 1

Let $P(x,y)$ be the point which moves such that sum of its distances from $F(3,0)$ and $F^{\prime }(-3,0)$ is $9$

The curve traced by $P$ is an ellipse length of whose major axis $(2a=9)$ and whose central is at the modpoint of$F{F}^{\prime }$ at $(0,0)$ where $F,F^{\prime }$ are the foci.

The major axis is the x-axis since the foci lie on it.

Now,

$CF=ae=3$

$a=\frac{9}{2}$

$\therefore e=\frac{3}{a}=\frac{3\times 2}{9}=\frac{2}{3}$

$b=a\sqrt{1-e^2}$

$b=\frac{9}{2}\sqrt{1-\frac{4}{9}}=\frac{3\sqrt{5}}{2}$

$b^2=\frac{45}{4}$

So, the equation is,

$\frac{x^2}{81/4}+\frac{y^2}{45/5}=1$