Submitted by Surfer Rosa on Sat, 09/30/2017 - 20:05

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.

Please help!

## 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.