Let A ( α , 0 ) and B ( 0 , β ) be the points on the line 5 x + 7 y = 50 . Let the point P divide the line segment A B internally in the ratio 7 : 3 . Let 3 x - 25 = 0 be a directrix of the ellipse E : x 2 a 2 + y 2 b 2 = 1 and the corresponding focus be S . If from S , the perpendicular on the x - axis passes through P , then the length of the latus rectum of E is equal to

Step-by-step solution

Given: A ( α , 0 ) and B ( 0 , β ) lies on the line 5 x + 7 y = 50 . ⇒ 5 α + 7 0 = 50 , 5 0 + 7 β = 50 ⇒ α = 10 , β = 50 7 Also, P divide the line segment A B internally in the ratio 7 : 3 . ⇒ P ≡ 7 × 0 + 3 × 10 7 + 3 , 7 × 50 7 + 3 × 0 7 + 3 ⇒ P ≡ 3 , 5 ⇒ a e = 3 as perpendicular from S passes through P Now, 3 x - 25 = 0 is a directrix of the ellipse E : x 2 a 2 + y 2 b 2 = 1 ⇒ x = 25 3 We know that, directrix of an ellipse is given by x = ± a e ⇒ a e = 25 3 Also, a e = 3 ⇒ a = 5 , b = 4 Length of LR = 2 b 2 a = 3