Area Of The Region Bounded By The Curves  Concepts & Shortcuts

Statutory warning : It's always best to sketch the curves than mugging up the formulae.
Some useful formulae:
Area bounded by the curves ax +/ m  = p and by +/ n = q is 4pq/ab sq units
Area bounded by ax +/ m + by +/ n = k is 2k^2/(ab)
Area bounded by ax + by = k and ax  by = k is 2k^2/ab
Area bounded by ax + by + ax  by = k is k^2/(ab)We will see in detail how we got these fancy formulas and how easy it is to derive them whenever you need it!
Area of a square with diagonal d = d^2/2
Area of a rhombus with diagonal d1 and d2 = (d1 x d2)/2
Area of parallelogram = base x heightSome important graphs
x = k (say k = 2) will give 2 lines parallel to y axis. One of which represents x = 2 and the other x =  2
Similarly y = k will give 2 lines parallel to x axis. One of which represents y = 2 and the other y =  2So x = 2 AND y = 2 we will give us a square with side = 2k as shown below.
Now what about x + 3 = 2 AND y + 1 = 2 ?
x + 3 = 2 => x = 1 or x = 5 (4 units)
y + 1 = 2 => y = 1 or y = 3 ( 4 units)
So this should also give a square of side 4 unitsGraph is as below
One interesting thing here is even if we plot x + 6 = 2 and y + 8 = 2
x + 6 = 2 => x = 8 or x = 4 (4 units)
y + 8 = 2 => y = 6 or y = 10 (4 units)
It will still give a square of side 4 units.Or say if we plot x  3 = 2 and y + 4 = 2
x  3 = 2 => x = 1 or x = 5 (4 units)
y + 8 = 2 => y = 6 or y = 10 (4 units)
again, a square of side 4 units.So all the values of x +/ a = k and y +/ b = k will yield a square of side 2k units. (definitely in different coordinates depending on a and b)
What if the value of k is different. Like x + 3 = 1 and y  1 = 2 ?
x + 3 = 1 => x = 2 or x = 4 (2 units)
y  1 = 2 => y = 3 or y = 1 (4 units)
A rectangle with sides 2 and 4.
We get a rectangle instead of square. That's it. Graph will be likeWe can say in general ax +/ m  = p and by +/ n = q will plot a rectangle with side 2p/a and 2q/b.
Hence area = 4pq/ab.Now we will see another important type.
x + y = 2 will give a square with diagonal = 2k as shown below.
Here also if you plot any x +/ a + y +/ b = 2 it will still yield a square with diagonal = 2k.
So in general, x +/ a + y +/ b = k will yield a square of diagonal 2k.
What about 3x + 4y = 12 ?
Graph is as below
here it is a rhombus with diagonals 6 and 8 (which is nothing but 12 x 2/3 and 12 x 2/4, where 3 and 4 are our coefficients). Area here is (2 x 12^2)/(3x4)
Here also if you try something like 3x + 6 + 4y  4 = 12, it will plot the same shape (a rhombus with diagonals as 6 and 8 and the only change will be the position of the rhombus in the xy plane which won't alter the area)
So in general, Area of ax +/ m + by +/ n = k is 2k^2/ab
We saw the graph of x + y = k, what about the graph of x  y = k ?
For example graph of x  y = 3 is as below
means it won't bound any region in the xy plane.
what about x  y = k type ?
For example, graph of x  y = 3 looks like
so this one also won't bound any region in the xy plane (good, lesser formulas!)
x + y = k case is also same. for example x + y = 3 won't bound any region in the xy plane. Graph is as below
What if we combine both ? i.e x + y = 3 and x  y = 3. Can you guess from the above graphs how it would turn out ?
Bounded region is as below
A square with diagonal as 6.
For example, the graph of 3x + 2y = 6 and 3x  2y = 6 is shown as below
So in general, if we plot ax + by = k and ax  by = k, we will get a rhombus with diagonals 2k/a and 2k/b.
Okay. so now we can solve the graph for x + y = 4 and x  y = 4. But what about x + y + x  y = 4 ?
Graph is as below
We can see that the graph will give a square of side = k
so area = k^2 = 16what would be the graph of 2x + y + 2x  y = 8 ?
Graph will be like
Rectangle with side 8 and 4. Area = 8 x 4 = 32
If you see this, the sides are nothing but 8/2 and 8/1. Where 2 and 1 are nothing by coefficients of x and y
So we can say that the area covered by the graph ax + by + ax  by = k is k/a * k/b = k^2/ab
How to plot the graph of a line, say 3x + 2y = 6 ?
when x = 0, y = 3
when y = 0, x = 2Graph is as below
x^2 + y^2 = r^2 is the equation of a circle with radius = r and center at origin.
For example, x^2 + y^2 = 9 will plot a circle as below
Share the formulas/concepts which we missed out and point out errors (if any).