Answer the following by appropriately matching the lists based on the information given in the paragraph Let the circles C1:x2+y2=9 and C2:(x−3)2+(y−4)2=16, intersect at the points X and Y. Suppose that another circle C3:(x−h)2+(y−k)2=r2 satisfies the following conditions : (i) centre of C3 is collinear with the centres of C1 and C2 (ii) C1 and C2 both lie inside C3, and (iii) C3 touches C1 at M and C2 at N. Let the line through X and Y intersect C3 at Z and W, and let a common tangent of C1 and C3 be a tangent to the parabola x2=8αy. There are some expression given in the List-I whose values are given in List-II below:
List-I
List-II
(I)
2h+k
(P)
6
(II)
Length of XY Length of ZW
(Q)
6
(III)
Area of triangle MZN
(R)
45
(IV)
α
(S)
521
(T)
26
(U)
310
Which of the following is the only INCORRECT combination ?
(A)(IV), (S)✓ Correct
(B)(IV), (U)
(C)(III), (R)
(D)(I), (P)
Answer: (A)
Solution
The centres C1(0,0) and C2(3,4) are 5 apart, with radii 3 and 4. Since C3 contains both and touches each, its diameter is 3+5+4=12, so r=6; its centre lies on line C1C2 at distance 3 from C1, namely (59,512), giving 2h+k=6. The common chord XY is 3x+4y−9=0; the chord ZW it cuts on C3 has length 5246 against XY=524, a ratio 6. The area of triangle MZN is 45, and the common tangent 3x+4y+15=0 of C1 and C3, made tangent to x2=8αy, gives α=310. So α pairs with 310, and pairing α with 521 is the incorrect combination.
JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Single correct option
