Q.449. Area of a quadrilateral.

Question 449.
In quadrilateral ABCD it's known that:
AC * BD = x and AB² - BC² + CD² - DA² = y
Find area of quadrilateral in terms of x and y.

Answer 449.
Refer to the following figure.

Let A(0, 0) be at the origin.
Let X-axis be along AC and C = (m, 0)
Let B = (a, b) and D = (c, d)

AC * BD = x
=> m * BD = x
=> BD^2 = x^2/m^2
=> (a - c)^2 + (b - d)^2 = x^2/m^2 ... ( 1 )

AB^2 - BC^2 + CD^2 - DA^2 = y
=> a^2 + b^2 - (m - a)^2 - b^2 + (m - c)^2 + d^2 - c^2 - d^2 = y
=> 2m (a - c) = y

Plugging (a - c) = y/2m in ( 1 ),
y^2/4m^2 + (b - d)^2 = x^2/m^2
=> l b - d l = √[x^2/m^2 - y^2/4m^2]

Area of the quadrilateral
= (1/2) AC * l b - d l
= (1/2) m * √[x^2/m^2 - y^2/4m^2]
= (1/2) √[x^2 - y^2/4)].

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