Question

 

Find the locus of the centroid of an equilateral triangle inscribed in the ellipse     

                  

Solution

Correct option is

 

Let the vertices of the equilateral triangle PQ and R and whose eccentric angles are α, β and γ.                   

Let the centroid of ∆PQR be (hk) then 

       

and, 

         

∵ ∆PQR is equilateral. 

∴ Centroid of the ∆PQR is same as the circumcentre. 

∵ Circumcentre of ∆PQR be 

       

       

Using (i) and (ii) then 

         

Squaring and adding (iii) and (iv) we get 

Hence locus of (hk) is 

        .

SIMILAR QUESTIONS

Q1

Find the equation of the ellipse referred to its centre whose minor axis is equal to the distance between the foci and whose latus rectum is 10.  

Q2

The extremities of a line segment of length l move in two fixed perpendicular straights lines. Find the locus of that point which divides this line segment in ratio 1 : 2.    

Q3

Find the lengths and equations of the focal radii drawn from the point  on the ellipse 25x2 + 16y2 = 1600.

Q4

 

For what value of λ dose the line y = x + λ touches the ellipse

9x2 + 16y2 = 144.

Q5

Find the equations of the tangents to the ellipse  which are perpendicular to the line y + 2x = 4.

Q6

Find the locus of the foot of the perpendicular drawn from centre upon any tangent to the ellipse .

Q7

Find the locus of the points of the intersection of tangents to ellipse  which make an angle θ.

Q8

Find the locus of the poles of tangents to  with respect to the concentric ellipse .  

Q9

 

Determine the equation of major and minor axes of the ellipse  

       

Also, find its centre, length of the latusrectum and eccentricity.

Q10

If SY and S1Y1 be perpendiculars from the foci upon the tangent atP of an ellipse, then Y and Y1 lie on the auxiliary circle andSY.S1Y1 =