Question

Find the area bounded by the lines y = 4x + 5, y = 5 – x and 4y = x + 5.

Answer 1

Given that y = 4x + 5  .....(i)

y = 5 – x  ......(ii)

And 4y = x + 5  ......(iii)

x	0	`-5/4`
y	5	0

 

x	0	5
y	5	0

 

x	0	–5
y	`5/4`	0

Solving equations (i) and (ii)

We get 4x + 5 = 5 – x

⇒ x = 0 and y = 5

∴ Coordinates of A = (0, 5)

Solving equations (ii) and (iii)

y = 5 – x

4y = x + 5

5y = 10

∴ y = 2 and x = 3

∴ Coordinates of B = (3, 2)

Solving equations (i) and (iii)

y = 4x + 5

4y = x + 5

⇒ 4(4x + 5) = x + 5

⇒ 16x + 20 = x + 5

⇒ 15x = – 15

∴ x = –1 and y = 1

∴ Coordinates of C = (–1, 1).

∴ Area of required regions = `int_(-1)^0 y_"AC" "d"x + int_0^3 y_"AB" "d"x - int_(-1)^3 y_"CB" "d"x`

= `int_(-1)^0 (4x + 5) "d"x + int_0^3 (5 - x) "d"x - int_(-1)^3 (x + 5)/4 "d"x`

= `[4  x^2/2 + 5x]_-1^0 + [5x - x^2/2]_0^3 - 1/4[x^2/2 + 5x]_-1^3`

= `[(0 + 0) - (2 - 5)] + [(15 - 9/2) - (0 - 0)] - 1/4[(9/2 + 15) - (1/2 - 5)]`

= `3 + 21/2 - 1/4[39/2 + 9/2]`

= `3 + 21/2 - 1/4 xx 24`

⇒ `3 + 21/2 - 6`

= `15/2` sq.units

Hence, the required area =  `15/2` sq.units