Question

Straight lines `(x)/(a)+(y)/(b)=1, (x)/(b)+(y)/(a)=1,(x)/(a)+(y)/(b)=2 and (x)/(b)+(y)/(a)=2` form a rhombus of area ( in square units)
A. `(ab)/(|a^(2)-b^(2)|)`
B. `(ab)/(a^(2)+b^(2))`
C. `(a^(2)b^(2))/(a^(2)+b^(2))`
D. `(a^(2)b^(2))/(|a^(2)-b^(2)|)`

Answer 1

Correct Answer - D
The equations of the four sides are
`(x)/(a)+(y)/(b)=1" "…(i)" "(x)/(b)+(y)/(a)=1`
`(x)/(a)+(y)/(b)=2" "…(iii) " "(x)/(b)+(y)/(a)=2" "…(iv)`
Clearly , (i) , (iii) and (ii), (iv) form two sets of parallel lines. So, the four lines form a parallelogram.
Area of rhombus `=|((2-1)(2-1))/({:((1)/(a),(1)/(b)),((1)/(b),(1)/(a)):})|=|(1)/((1)/(a^(2))-(1)/(b^(2)))|`
`implies` Area of the rhombus `=(a^(2)b^(2))/(|b^(2)-a^(2)|)`