computerbot wrote:
Two questions:
1.
GreenlightTestPrep wrote:
We can determine that the equation of the line is: y = -x
So, every point ON the line satisfies the equation y = -x
Also notice that the line divides the xy plane into TWO regions.
All points in the TOP region are such that y > -x
And all points in the BOTTOM region are such that y < -x
Is the fact that y>-x for TOP region and y<-x for BOTTOM region only true for line equations y = -x or is it true for ALL lines in the XY plane?
For every line in a 2D plane ax + by + c = 0. All the points in the 2D plane can be covered using the following conditions:
ax + by +c > 0 for all points above the line
ax + by + c = 0 for all the points on the line
ax + by +c < 0 for all points below the lineSo, y>-x for TOP region and y<-x for BOTTOM region is true only for the line y = -x, but for other lines you can use the concept explained above.
2.
GreenlightTestPrep wrote:
The point (p, q) in the BOTTOM region, which means q < -p
If we add p to both sides, we get: p + q < 0
In other words, (p + q) is some NEGATIVE value
computerbot wrote:
Two questions:
If we didn't have the line with equation y = -x (and had some other equation say y = 3x-2 or y = -3x - 2), would the above boldfaced portion still be true?
If we did not have y = -x line and had let's say y = 3x-2 ( or y - 3x + 2 = 0) then the conditions will change as follows
y - 3x + 2 > 0 for all points above the line
y - 3x + 2 = 0 for all the points on the line
y - 3x + 2 < 0 for all points below the line
And if you substitute the point (p, q) in it you will get
q - 3p + 2 < 0 (As the point is below the line)
And for (a, b) you will get
b - 3a + 2 > 0 as the point is above the line.
Hope it helps!