# Area of a Graphs

Before we proceed ahead with calculating the area of the combination of graphs, we should be clear with the quadrants and the signs of X and Y in the same.

To find the area of graphs, we first need to sketch the graphs of the equations and then by using geometry/ coordinate geometry we can find the area of the enclosed figure.

Example-1

Find out the area of the region enclosed by

*y*= |*x*| and*y*= 2.Solution

Following is the area enclosed by the equations given above:

Example-2

In the X â€“ Y plane, the area of the region bounded by the graph |

*x*+*y*| + |*x*âˆ’*y*| = 4 is- 8
- 12
- 16
- 20

Solution

Let

Then |

â‡’

Similarly,

The area in the first quadrant is 4.
By using symmetry, the total area in all the four quadrants = 4 Ã— 4 = 16 sq. units

*x*â‰¥ 0,*y*â‰¥ 0 and*x*â‰¥*y*Then |

*x*+*y*| + |*x*âˆ’*y*| = 4â‡’

*x*+*y*+*x*â€“*y*= 4 â‡’*x*= 2Similarly,

*x*â‰¥ 0,*y*â‰¥ 0,*x*â‰¤*y**x*+*y*+*y*â€“*x*= 4 â‡’*y*= 2The area in the first quadrant is 4.

Example-3

If

*p, q*, and*r*are any real numbers, then- max (p, q) < max (p, q, r)
- min (p, q) = (p + q âˆ’|p âˆ’ q|)
- min (p, q) < min (p, q, r)
- None of these

Solution

If we take
(

Similarly, If
(

So, option (b) is the answer.

*r*<*p, q*, then (1) and (3) cannot hold. For (2), if*p*â‰¥*q*, then |*p*âˆ’*q*| =*p*â€“*q*.*p + q*âˆ’*|p*âˆ’*q|*) = (*p + q*âˆ’*p*+*q|*) =*q*= min (*p, q*)Similarly, If

*p*<*q*then |*p*â€“*q*| =*q*â€“*p*.*p*+*q*âˆ’ |*p*âˆ’*q*|) = (*p + q*âˆ’*q*+*p*) =*p*= min (*p, q*)So, option (b) is the answer.

Example-4

If

*a b c d*= 1,*a*> 0,*b*> 0,*c*> 0,*d*> 0, then what is the minimum value of (*a*+ 1) (*b*+ 1) (*c*+ 1) (*d*+ 1)?- 1
- 8
- 16
- None of these

Solution

The minimum value will occur when
So, the minimum value of (

*a*=*b*=*c*=*d*= 1*a*+ 1) (*b*+ 1) (*c*+ 1) (*d*+ 1) = 16Example-5

Consider a triangle drawn on the X-Y plane with its three vertices (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X,Y) coordinates. What is the number of points with integer coordinates inside the triangle (excluding all the points on the boundary)?

- 780
- 800
- 820
- 741

Solution

The equation formed from the data is
The values which will satisfy this equation are

(1, 39), (1, 38) â€¦(1,1)

(2, 38), (2, 37), â€¦(2,1)

(39, 1)
So the total number of cases are 39 + 38 + 37 â€¦ + 1

*x*+*y*< 41(1, 39), (1, 38) â€¦(1,1)

(2, 38), (2, 37), â€¦(2,1)

(39, 1)

Example-6

A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs 250 and Rs 300 per day respectively. In addition, a male operator gets Rs 15 per call he answers and female operator gets Rs 10 per call she answers. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?

- 15
- 14
- 12
- 10

Solution

First let us form both the equations:

40 m + 50 f = 1000

250 m + 300 f + 40 Ã— 15 m + 50 Ã— 10 Ã— f = A

850 m + 8000 f = A
Where m and f are the number of males and females and A is the amount paid by the service provider.
Then the possible values for f are 8, 9, 10, 11, 12
If f = 8, then m = 15
If f = 9, 10 and 11 then m will not be an integer while

f = 12 then m will be 10.
By putting f = 8 m = 15 and A = 18800. When f = 12 and m = 10 then A = 18100.
Hence, the number of males will be 10.

40 m + 50 f = 1000

250 m + 300 f + 40 Ã— 15 m + 50 Ã— 10 Ã— f = A

850 m + 8000 f = A

f = 12 then m will be 10.

Example-7

If

*a*,*b*and*c*are the sides of a triangle, then what is the maximum value of the expression?- 1
- 3/2
- 2
- 5/2

Solution

Assume 2s =
Hence,
Similarly,
Hence,

*a*+*b*+*c*. We know that*b*+*c*>*a*, so we get 2(*b*+*c*) >*a*+*b*+*c*= 2s*b*+*c*> s*c*+*a*> s,*a*+*b*> sExample-8

At how many distinct points the graphs of

*y*=*x*^{-}^{1}and*y*= log_{e}*x*intersect? (CAT 2003)Solution

We can see that the graphs of

*y*=*x*^{-}^{1}and*y*= log_{e}*x*intersect just once.Example-9

How many integral solution is/are possible for the equation |

*y*âˆ’18| + |*y*â€“ 9| + |*y*+9| + |*y*+ 18| = 54?Solution

Here, |
Hence |
Now, for any point
So, for point
For points outside this limitation, this expression will have different values.
Hence, the required numbers are â€“9, âˆ’8,â€¦, 8, 9. So, there will be a total of 19 values.

*y*âˆ’*N*| should be seen as nothing but the distance of the point*y*from the point*N*on the number line, a person standing at a point*N*.*y*âˆ’ 18| + |*y*â€“ 9| + |*y*+ 9| + |*y*+ 18| is the sum of the distances of the point*y*from 18, 9, âˆ’9 and â€“18.*y*, where*p*â‰¤*y*â‰¤*q*, the sum of the distances from*p*and*q*is*q*â€“*p*.*y*, where âˆ’9 â‰¤*y*â‰¤ 9, the sum of the distances of*y*from â€“18, âˆ’9, 9 and 18 is [18âˆ’(âˆ’18)] + [9âˆ’ (âˆ’9)] = 54