Points on either side of a line

[mmaismma]

Summary:: The set of values of $b$ for which the origin and the point $(1, 1)$ lie on the same side of the straight line $a^2x+aby+1=0$ $\forall~a\in\mathbb{R},~b>0$.(a) $a\geq1$ or $a\leq-3$
(b) $a\in~(-3,~0)\cup(\frac13,~1)$
(c) $a\in~(0,~1)$
(d) $a\in~(-\infty,~0)$

I tried solving it but I didn't get an answer:

$f(x)=a^2+aby+1=0\\ f(0, 0)=0+0+1>0\\ So,~f(1, 1)>0\\ a^2+ab+1>0\\ {}^A/_Q~a\in\mathbb{R}\\ So,~D>0\\ b^2-4.1>0\\ b^2>4\\ b\in(2,~\infty)$

 

[Svein]

Hint: As long as the intersection between the line given and the line through (0, 0) and (1, 1) is outside the stub between (0, 0) and (1, 1), both points lie on the same side of the given line. So, find the intersection...

 

[mfb]

I moved the thread to our homework section.
The problem statement talks about the set of values for b but the multiple choice answers are about a. What is it?

While the approach of post 2 is possible I like the approach of post 1 more.

 

[SammyS]

Summary:: The set of values of $b$ for which the origin and the point $(1, 1)$ lie on the same side of the straight line $a^2x+aby+1=0$ $\forall~a\in\mathbb{R},~b>0$.(a) $a\geq1$ or $a\leq-3$
(b) $a\in~(-3,~0)\cup(\frac13,~1)$
(c) $a\in~(0,~1)$
(d) $a\in~(-\infty,~0)$

I noticed that the title of this tread tells us that the points are on either side of the line, meaning to me that the line passes between the points.

On the other hand, your statement of the problem says that the points, (0, 0) and (1, 1) both lie on the same side of the line.

Looking at the choices given for the answer, I suspect that the Title gives the correct version. That also agrees with your result for $b$.

Edit:
Another thought. No matter which of the two possibilities is being asked, consider the following.

It's straight forward to determine each of the following in terms of $a$ and $b$: the slope and the x and y intercepts. Furthermore, the x intercept depends only on $a$, not on $b$. In fact the x intercept is negative for all allowed values of $a$.

With this in mind, you can determine the parameters needed for the line to have a negative x intercept and which also passes through (1, 1). This line will separate those lines passing between the two points from those for which both points are on the same side.