AP EAMCET201922 Apr 2019Morning ShiftMathematicsEllipseActual
If ((l, m) ) is the circumcentre of an equilateral triangle inscribed in the ellipse ( x^2 a^2 + y^2 b^2 =1 ) having vertices at points with eccentric angles ( ₁, ₂ ) and ( ₃ ), then ( 2 3 [ ( ₁- ₂ )+ ( ₂- ₃ )+ ( ₃- ₁ ) ]= )
Options
- A( 9 l^2 2 a^2 + 9 m^2 b^2 -1 )
- B( l^2 a ^2 + m^2 b^2 -3 )
- C( 3 l^2 a^2 + 3 m^2 b^2 -1 )
- D( 3 l^2 a^2 + 3 m^2 b^2 - 3 2 )
Correct answer
C. ( 3 l^2 a^2 + 3 m^2 b^2 -1 )
Step-by-step solution
As we know coordinates of circumcentre of an equilateral triangle is same as centroid, so ( aligned & (l, m)= ( a ₁+a ₂+a ₃ 3 , b ₁+b ₂+b ₃ 3 ) & 3 l a = ₁+ ₂+ ₃ (i) & and 3 m b = ₁+ ₂+ ₃ (ii) aligned ) Now, after squaring and adding the Eqs. (i) and (ii), we get ( aligned & 9 l^2 a^2 + 9 m^2 b^2 = ^2 ₁+2 ₁ ₂+ ^2 ₁ & 9 l^2 a^2 + 9 m^2 b^2 =3+2 ₁ ₂ & 2 3 ( ₁- ₂ )= 3 l^2 a^2 + 3 m^2 b^2 -1 & 2 3 [ ( ₁- ₂ )+ ( ₂- ₃ )+ ( ₃- ₁ ) ] & = 3 l^2 a^2 + 3 m^2 b^2 -1 aligned )