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Consider an ellipse E: x^(2)/a^(2)+y^(2)...

Consider an ellipse E: `x^(2)/a^(2)+y^(2)/b^(2)=1`, centered at point 'O' and having AB and CD as its major and minor axes respectively if `S_1`be one of the focus of the ellipse, radius of the incircle of ∆`OCS_1` be 1 unit and ` OS_1=6` units.
Q. If perimeter of `∆OCS_1` is p units, then the value of p is

A

`65pi//4`

B

`64pi//5`

C

`64pi`

D

`65pi`

Text Solution

Verified by Experts


`:.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)`
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