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KVPY2010MathematicsEllipse

Let P be an arbitrary point on the ellipse x ² a ² + y ² ~b ² -1, a > b >0 . Suppose F ₁ and F ₂ are the foci of the ellipse. The locus of the centroid of the triangle PF ₁ ~F ₂ as P moves on the ellipse is-

Options

  1. Aa circle
  2. Ba parabola
  3. Can ellipse
  4. Da hyperbola

Correct answer

C. an ellipse

Step-by-step solution

array l P a , b G ( x _ i 3 , y _ i 3 ) F ₁ ( ae , 0) F ₂ (- ae , 0) h = acos +a e - ae 3 ; = 3 ~h a array array l k= b 3 ; = 3 k b ² + ² =1 9 h² a² + 9 k² b² =1 x² (a² / 9 ) + y² (b² / 9 ) =1 (Ellipse) array

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