In which of the following cases does S represent a pair of distinct straight lines?

Question

Let $S=\{(x,y)∈R^2 \; | \; ax^2 + 2hxy +by^2=0 \}$.

In which of the following cases does S represent a pair of distinct straight lines

$$\begin{array}{cl} a) & a=b=5,h=-1\\ b) & a=b=3,h=4 \\ c) & a= 1,b=4, h=2 \\ d) & \text{None of the above.}\end{array}$$

My attempts :

When $h^2-ab <0$, S represents an ellipse : so option a) is ellipse

When $h^2- ab> 0$, S represents a hypberbola : so option b) is hyperbola

Option c) $ h^2= ab$, S represents a parabola

So the correct answer is none of these, i.e. option D)

Is my answer is correct or not?

Answer 1

You will get ellipses, hyperbolas etc if the right hand side is not zero. In this case,

• option (a) is $$\eqalign{ 5x^2-2xy+5y^2=0\quad &\Leftrightarrow\quad (5x-y)^2+24y^2=0\cr &\Leftrightarrow\quad 5x-y=0\ ,\quad y=0\cr &\Leftrightarrow\quad x=y=0\cr}$$ which is a point;
• option (b) is $$\eqalign{ 3x^2+8xy+3y^2=0\quad &\Leftrightarrow\quad (3x+4y)^2=7y^2\cr &\Leftrightarrow\quad 3x+(4-\sqrt7)y=0\quad\hbox{or}\quad 3x+(4+\sqrt7)y=0\cr}$$ which is a pair of straight lines;
• option (c) is $$\eqalign{ x^2+4xy+4y^2=0\quad &\Leftrightarrow\quad (x+2y)^2=0\cr &\Leftrightarrow\quad x+2y=0\cr}$$ which is one straight line.

Answer 2

Your option $(b)$ could as well turn into a pair of straight lines.

Substitute $y=kx$ in option $(b)$ to get $$(3k^2+8k+3)x^2=0$$ Upon solving $$(3k^2+8k+3)=0$$ for $k$ we get two real solutions $$k_1=\frac {-4+\sqrt 7}{3}\text { and } k_2=\frac {-4-\sqrt 7}{3}$$

Thus option (b) sounds correct.