premise and function analysis

consider the following three possible conditions on two real numbers x and y

$$p : x$$ and $$y$$ satisfy the equation $$(x+y)^2 = a(x^2+y^2) + bxy$$ where $$a,b$$ are real constants.

$$q : x = 0$$ and $$y = 0$$

$$r : x = 0$$ or $$y=0$$

1. suppose that in condition $$p, a = b = 1$$ then $$p$$ is [H] for $$q$$ and $$p$$ is [I] for $$r$$.
2. suppose that in condition $$p, a = b = 2$$ then $$p$$ is [J] for $$q$$ and $$p$$ is [K] for $$r$$.
3. if in condition $$p$$ we set $$a=2$$ we can transform the equation in $$p$$ into $${(x + \frac{b - [L]}{[M]}y)}^2 + ([N] - \frac{{(b-[O])}^2}{[P]})y^2 = 0$$

hence $$p$$ is a necessary and sufficient for$$q$$ if and only if $$b$$ satisfies $$[Q] < b < [R]$$

we are to find $$[H]$$ until $$[R]$$

for $$[H], [I], [J],[K]$$ the are options to choose

$$0$$ necessary and sufficient condition

$$1$$ a necessary condition but not sufficient condition

$$2$$ a sufficient condition but not necessary condition

$$3$$ neither a necessary condition nor sufficient condition

i know some examples, for sufficient condition the example = boiling potato is a sufficient condition to cook it, but not necessary since there are many ways to cook potato. an example for necessary condition = if want one to go to college one must be a human, being a human is necessary condition to go to college but not sufficient. example for necessary and sufficient condition = for a building to be called a house is to have alive being to live it.

but how to apply it in these math equations.. any people who is good with this subjects? i am not good with this subject. or anyone can give me some hints where should i start?

• What does "p is [H] for q" mean? (By the way, you should read the description of a tag before applying it - this has nothing whatever to do with functional analysis...) – David C. Ullrich Feb 14 at 15:07