Let E 1 and E 2 be two ellipses whose centers are at the origin. The major axes of E 1 and E 2 lie along the x-axis and the y-axis, respectively. Let S be the circle x 2 + (y – 1) 2 = 2. The straight line x + y = 3 touches the curves S, E 1 and E 2 at P,Q and R, respectively. Suppose that PQ = PR = 
. If e 1 and e 2 are the eccentricities of E 1 and E 2 , respectively, then the correct expression(s) is (are)
+ 
= 
| = 
Text Solution
Verified by ExpertsA
(a.b)
Sol. E 1 → 
+ 
= 1
E 2 = 
= 1
Now as x + y = 3 is a tangent
a 2 + b 2 = A 2 + B 2 = 9
Now point P is
x 2 + (2 – x) 2 = 2
2x 2 – 4x + 2 = 0
x = 1
so P is (1, 2)
points Q & R are 
& 
Now 
lies on E 1 so 
= 1
⇒ 225 – 25a 2 + 16a 2 = 9a 2 (9– a 2 ) ⇒ 225 – 9a 2 = 9a 2 (9 – a 2 )
⇒ 25 – a 2 = a 2 (9 – a 2 )
⇒ a 4 – 10a 2 + 25 = 0 ⇒ a 2 = 5 so b 2 = 4
e 12 = 
Now 
lies on E 2
+ 
= 9
9 – A 2 + 64A 2 = 9A 2 (9 – A 2 )
1 + 7A 2 = A 2 = 9A 2 – A 4 ⇒ A 4 – 2A 2 + 1 = 0 ⇒ A 2 = 1 so B 2 = 8
e 22 = 
──────────────────────────────────────────────────────────────────────────────────────────
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems