Question

An ellipse `E,(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`, centred at point O has AB and CD as its major and minor axes, respectively. Let `S_(1)` be one of the foci of the ellipse, the radius of the incircle of traingle `OCS_(1)` be 1 unit, adn `OS_(1)=6` units
The area of ellipe (E) is
A. `65pi//4`
B. `64pi//5`
C. `64pi`
D. `65pi`

Answer 1

`:.OS_(1)=ae=6,OC=b`
Also,` CS_(1)=a`
`:. "Area of " DeltaOCS_(1)=(1)/(2)=(OS_(1))xx(OC)=3b`
`:.` Semi-perimeter of `DeltaOCS_(1)=(1)/(2)=(OS_(1)+OC+CS_(1))`
`=(1)/(2)(6+a+b) " "(1)`
`:.` In radius of `DeltaOCS_(1) ", "( "Using" r=(Delta)/(S))`
=` (3b)/((1)/(2)(6+a+b))=1" "("Using"r=(Delta)/(S))`
`or 5b=6+a" "(2)`
Also, `b^(2)=a^(2)-a^(2)e^(2)=a^(2)-36" "(3)`
From (2), we get
`25(a^(2)-36)=36+a^(2)+12a`
`or 2a^(2)-a-78=0`
`a=(13)/(2),-6`
`:. a=(13)/(2) and b=(5)/(2)`
Area or ellipse `=piab=(65pi)/(4)` sq. unit
Perimeter of `DeltaOCS_(1)=6+a+b=6+(13)/(2)+(5)/(2)=15` units
The equation of director circle is
`x^(2)+y^(2)=a^(2)+b^(2)`
or `x^(2)+y^(2)=(97)/(2)`