Question

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 =    

Solution

Correct option is

b2

 

Equation of tangent at is 

          

 

Equation of the perpendicular to (i) and passing through (ae, 0) is

      

For locus of Y, eliminate Ï• from equations (i) and (ii), for which squaring and adding (i) and (ii), we get 

      

                                                       

   

Hence locus of Y is the auxiliary circle. Similarly, as above we can prove that the locus of point Y1 is the auxiliary circle. 

                                     

                                 

                                 

                                  = b2.     

SIMILAR QUESTIONS

Q1

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.    

Q2

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

Q3

 

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

9x2 + 16y2 = 144.

Q4

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

Q5

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

Q6

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

Q7

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

Q8

 

Determine the equation of major and minor axes of the ellipse  

       

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

Q9

 

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

                  

Q10

Find the condition on a and b for which two distinct chords of the ellipse  passing through (a, –b) are bisected by the line x + y = b