Home
Class 12
MATHS
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)`
Promotional Banner

Similar Questions

Explore conceptually related problems

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. The equation of the director circle of (E) is

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 perimeter of DeltaOCS_(1) is

Consider an ellipse E ,(x^2)/(a^2)+(y^2)/(b^2)=1 , centered at point O andhaving A B and C D as its major and minor axes, respectively. If S_1 is one of the focus of the ellipse, the radius of the incircle of triangle O C S_1 is 1 unit, and O S_1=6 units, then the value of (a-b)/2 is_________

A tangent having slope of -4/3 to the ellipse (x^2)/(18)+(y^2)/(32)=1 intersects the major and minor axes at points A and B , respectively. If C is the center of the ellipse, then find area of triangle A B Cdot

The tangent at point P on the ellipse x^(2)/a^(2) + y^(2)/b^(2) = 1 cuts the minor axis in Q and PR is drawn perpendicular to the minor axis. If C is the centre of the ellipse, then CQ*CR =

Normal to the ellipse (x^2)/(64)+(y^2)/(49)=1 intersects the major and minor axes at Pa n dQ , respectively. Find the locus of the point dividing segment P Q in the ratio 2:1.

tangent drawn to the ellipse x^2/a^2+y^2/b^2=1 at point 'P' meets the coordinate axes at points A and B respectively.Locus of mid-point of segment AB is

The tangent to x^(2)//a^(2)+y^(2)//b^(2)=1 meets the major and minor axes in P and Q respectively, then a^(2)//CP^(2)+b^(2)//CQ^(2)=

If a tangent to the ellipse x^2/a^2+y^2/b^2=1 , whose centre is C, meets the major and the minor axes at P and Q respectively then a^2/(CP^2)+b^2/(CQ^2) is equal to

If normal at any point P to ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1(a > b) meet the x & y axes at A and B respectively. Such that (PA)/(PB) = 3/4 , then eccentricity of the ellipse is: