Assignment 1 - STEP Question - c^2 condition

Hi

I'm very stuck on the very last part of the STEP Question in Assignment 1.

I'm not sure what the question even means. There are so many ifs and only ifs and the reasoning seems to be running one way then the other. I can't follow it.

I've done everything up to this point including answering the part about "Show that". But I can't understand the last line. I'm so confused I don't even know what my question is. Anyway here's the relevant part of the question:

Show that this condition can be written
$$c^2 = 1− \left(\frac{a + b}{a-b}\right)^2$$
and deduce that it can only hold if $0 < c^2 \le 1$

Here's my reasoning for "deduce that it [what?] can hold only if $0 < c^2 \le 1$"

$c^2$ is positive or zero because all the numbers are real
$c^2$ can't be zero because then c = 0 and you have two distinct real roots from part (i)
$c^2 = 1-A^2$ with $A$ positive or zero so $c^2 \le 1$
If $a= -b$ then $A$ is zero otherwise it's positive this shows that $c^2 = 1$ is allowed

Taking these together and tidying up gives
$0 < c^2 \le 1$

But I feel like I should be starting from here not ending here.

Last part of 3ii

Actually, what you have written here looks pretty good! Personally I would start with showing that $0 \le c^2 \le 1$ (as it is a real number squared, and is also equal to $1-$ (a real number squared). I would then go on to show that $c=0$ is not possible, as then we have $(a-b)^2=(a+b)^2 \implies ab=0$, which implies that one of $a$ or $b$ (or both) is zero, which contradicts the information given in the stem,

Then finish by showing that $c$ can be equal to 1.

"Only if"

Many Thanks. I think my confusion was, and to be honest still is, with the "only if" terminology. I shall go away and contemplate. :-)

"only if"

If we do not have $0 \lt c^2 \leq 1$ then the condition cannot be true. The condition can only be true if $c^2$ is in the given range, Note that having $0 \lt c^2 \le 1$ does not mean that the condition MUST hold, $0 \lt c^2 \leq 1$ is a necessary condition, but not necessarily a sufficient one!

There is some more about "if" and "only if" in Assignment 10.

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