Finding parameter $k$ for which $\left(x-\frac{1}{2}\right)^2+\left(y+\frac{1}{2}\right)^2= \frac18k^2(x+y+1)^2$ is an equation of an ellipse

Question

I need to find the value of $k$ such that the following equation represents an ellipse:-

$$\left(x-\frac{1}{2}\right)^2+\left(y+\frac{1}{2}\right)^2= \frac18k^2(x+y+1)^2$$

I need to find the range of $k$. Is there any other way than expanding the RHS? Because that would be really tedious and quite a nightmare.

Answer 1

Note that ellipse can be characterized as the locus of points $P=(x,y)$ whose distances to a point $F$ and a straight line $d$ are in a fixed ratio less than 1: $$\frac{\text{dist}(P, F)}{\text{dist}(P, d)} = c < 1$$ (see also Certainlynotadog's comment above).

In your case, consider $F=(1/2,-1/2)$ and $d$ as $x+y+1=0$. Then the given equation becomes $$\text{dist}^2(P, F)=\frac{k^2}{4}\text{dist}^2(P, d)$$ and therefore the locus is an ellipse if and only if $k^2/4<1$ that is $|k|<2$. For $k=0$ we have the degenerate case where the ellipse is the point $F$.