Consider ellipses E k : k x 2 + k 2 y 2 = 1 , k = 1 , 2 , … , 20 . Let C k be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse E k . If r k is the radius of the circle C k , then the value of ∑ k = 1 20 1 r k 2 is

Step-by-step solution

Given, E k :   k x 2 + k 2 y 2 = 1 ⇒ E k   : x 2 1 k 2 + y 2 1 k 2 = 1 Now equation of the chord joining the points 1 k , 0   &   0 , 1 k will be, L k : x 1 k + y 1 k = 1 ⇒ k x + k y - 1 = 0 Now r k = Perpendicular distance of L k from ( 0 , 0 ) we get, r k = - 1 k + k 2 ⇒ r k 2 = 1 k + k 2 Now putting the value of r k 2 in ∑ k = 1 20 1 r k 2 we get, ∑ k = 1 20 1 r k 2 = ∑ k = 1 20 k + k 2 = 20 × 21 2 + 20 × 21 × 41 6 = 210 + 2870 = 3080