Question

One of the diameters of the circle circumscribing the rectangle ABCD is x – 4y + 7 = 0. If A and B are points (– 3, 4) and (5, 4) respectively, then the area of the rectangle is …….. 

(a) 32 sq. units

(b) 16 sq. units

(c) 64 sq. units

(d) 8 sq. units   

Answer 1

The correct option (a) 32 sq. units

Explanation:

Diameter is x – 4y + 7 = 0

point A(– 3, 4) & B(5, 4) does not lie on the diameter.

Let E be centre of circle.

∵ ABCD is a rectangle

|EA| = |EB| and E(a, b)

∴      (a + 3)² + (b – 4)² = (a – 5)² + (b – 4)²

∴        6a + 9 = – 10a + 25

∴          a = 1 

and E lie on line x – 4y + 7 = 0

⇒ a – 4b = 7 ⇒ b = 2

∴ E(a, b) = (1, 2) 

Now    |AB| = √[(5 + 3)² + (4 – 4)²] = 8

|BD| = 2|EB| 

|BD| = 2√[(5 – 1)² + (4 – 2)²] = 2√(20) = 4√5

From ΔABD,

AD² = BD² – AB²  = (4√5)² – (8)² 

= 80 – 64 

= 16

∴  |AD| = 4

∴ Area of rectangle ABCD  = |AB| |AD|

= (8)(4)

= 32 sq. units.