# Range of value to satisfy sufficient and necessary condition

1. Nov 1, 2015

### songoku

1. The problem statement, all variables and given/known data
Let
p: x and y satisfy inequality x2 + y2 + 4x - 8y + c < 0 where c is real number
q: x and y satisfy x - y + 8 > 0 ; 4x + 3y - 24 < 0 ; y < 0

Find the range of c so that:
a. p is sufficient condition for q
b. p is necessary condition for q

2. Relevant equations
Circle
Inequality

3. The attempt at a solution
p is the region inside circle (excluding circumference), centered at (-2, 4) and has radius of √(20 - c)

q is the infinite region bounded by two lines and x - axis

I want to ask whether my opinion is correct:
a. For p to be sufficient condition for q, region q should be located inside the circle
b. For p to be necessary condition for q, the circle should be inside the region of q

Thanks

2. Nov 1, 2015

### andrewkirk

I think it's the other way around.
'p is sufficient for q' means $p\Rightarrow q$ which should mean that $p\subseteq q$.

3. Nov 1, 2015

### HallsofIvy

Staff Emeritus
'p' is the interior of a circle with radius depending on c, 'q' is the region between two intersecting lines and below the y-axis . 'if p then q' (p is a sufficient condition for q) says that if (x,y) is in the circle it must be in the region between the lines- the circle must be a subset of the region. 'if q then p' (p is a necessary condition for q) says that if (x, y) is in the region it must be a in the circle- the region is a subset of the circle. But since the region between the line extends infinitely, that does not see, possible for any 'c'. If the last condition were "y> 0", then it would be a bounded triangular region.

4. Nov 1, 2015

### songoku

I get it.

Thanks a lot for all the help