Question

Rectangle $ABCD$ has area $200$ sq. unit. An ellipse with area $200\pi $ sq. unit passes through $A$ and $C$ has focii at $B$ and $D$ . Then perimeter of the rectangle $ABCD$ is

• A. $80$ unit
• B. $40$ unit
• C. $20$ unit
• D. $60$ unit

Answer 1

Answer: (A)

Area of ellipse $\pi ab=200\pi $ sq. unit
$\Rightarrow ab=200\ldots \left(i\right)$
Given $AB\times AD=200\ldots \left(ii\right)$
$AB+AD=2a$
Squaring both sides
$AB^{2}+AD^{2}+2AB\times AD=4a^{2}$
$BD^{2}+2\times 200=4a^{2}$
$4a^{2}e^{2}+400=4a^{2}$
$100=a^{2}-a^{2}e^{2}$
$100=b^{2}$
$\Rightarrow b=10$ unit
$\therefore ab=200$
$a=20$ unit
Perimeter of rectangle
$ABCD=2\left(AB + AD\right)$
$=2\times 2a$
$=4a$
$=4\times 20=80$ unit