Question

The quadrilateral formed by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$ and $y=bx+d$ has area $18$. The quadrilateral formed by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$ and $y=bx-d$ has area $72$. If $a,b,c,d$ are positive integers then the least possible value of the sum $a+b+c+d$ is

• A. 13
• B. 14
• C. 15
• D. 16

Answer 1

The correct option is D

16

Smaller Quadrilateral

Area = 2 [($\Delta$ ABC)]

$=2\left(\frac{1}{2}\frac{|d-c||d-c|}{|a-b|}\right)$

$=\frac{(c-d{)}^2}{|a-b|}=18$ . . . . (1)

Similarly, for larger Quadrilateral

$\text{Area}=\frac{(c+d{)}^2}{|a-b|}=72$ . . .(2)

Dividing (1) & (2), we get

c= 3d

$\therefore \frac{d^2}{|a-b|}=\frac{9}{2}$

For $a+b+c+d$ to be minimum & be positive integers, only possible values are $a=3,b=1,c=9,d=3$ or $a=1,b=3,c=9,d=3$.