Characterizing Conic Section By Eigenvalues

Question

We can bring a function of the form $$ax^2+2bxy+cy^2+dx+ey+f=0$$

To the form of $$\lambda_{1}(x'')^2+\lambda_{2}(y'')^2+k=0$$ In a textbook It is written that if:

1. $\lambda_{1}\lambda_{2}>0$ and the sign of $k$ is opposite to this of $\lambda_{1}$ it is an ellipse

2. $\lambda_{1}\lambda_{2}>0$ and the sign of $k$ is same to this of $\lambda_{1}$ it is an empty set

3. $\lambda_{1}\lambda_{2}>0$ and $k=0$ it is a point

4. $\lambda_{1}\lambda_{2}>0$ and $k\neq 0$ it is an hyperbola

5. $\lambda_{1}\lambda_{2}>0$ and $k=0$ it is two straight lines

But (1) can be (4) and (5) can be (3)

Answer 1

As serg_1 notes, two of the relation signs are not OK. Correct statements are:

1. $\lambda_{1}\lambda_{2}>0$ and the sign of $k$ is opposite to this of $\lambda_{1}$ it is an ellipse

2. $\lambda_{1}\lambda_{2}>0$ and the sign of $k$ is same to this of $\lambda_{1}$ it is an empty set

3. $\lambda_{1}\lambda_{2}>0$ and $k=0$ it is a point

4. $\lambda_{1}\lambda_{2}<0$[** N.B. "$<$"] and $k\neq 0$ it is an hyperbola

5. $\lambda_{1}\lambda_{2}<0$[** N.B. "$<$"] and $k=0$ it is two straight lines