If a x^2+b y^2=15 is the equation of the ellipse for which distance between its foci is 2 and distance between its directrices is 5 , then a+b=

Step-by-step solution

Given equation is a x^2+b y^2=15 x^2 ( 15 a )^2 + y^2 ( 15 b )^2 =1 Now, a^ = 15 a , b^ = 15 b According to question, aligned & 2 d e=2 a^ e=1 & 2 a^ e =5 a^ e = 5 2 aligned Multiply (ii) and (iii), a^ e a^ e =1 5 2 a^ 2 = 5 2 a^ = 5 2 from (i) a^ e=1 aligned & e= 2 5 & b^ 2 =a^ 2 = (1-e^2 ) & b^ 2 = 5 2 (1- 2 5 )= 5 2 3 5 = 3 2 & b^ = 3 2 aligned Now, compare a^ & b^ values, 15 a = 5 2 square both sides, 15 a = 5 2 a=6 . Similarly, 15 b = 3 2 aligned & 15 b = 3 2 b=10 & So, a+b=10+6=16 aligned