Let the circle `C_(1):x^(2)+y^(2)=9` and `C_(2):(x-3)^(2)+(y-4)^(2)=16` intersect at the point X and Y. Suppose that another circle `C_(3):(x-h)^(2)+(y-k)^(2)=r^(2)` satisfies the following conditions
(i). Centre of `C_(3)` is collinear with the center of `C_(1)&C_(2)`
(ii). `C_(1)&C_(2)` both lie inside `C_(3)` and
(iii). `C_(3)` touches `C_(1)` at M and `C_(2)` at N
Let hte line through X and Y intersect `C_(3)` at Z and W and let a common tangent of `C_(1)` & `C_(3)` be a tangent to the parabola `x^(2)=8alphay`
There are some expressions given in the following lists
`{:("List I","List II"),((I)" "2h+k,(P)" "6),((II)" "("length of ZW")/("length of XY"),(Q)" "sqrt(6)),((III)" "("Area of "DeltaMZN)/("Area of "DeltaZMW),(R)" "(5)/(4)),((IV)" "alpha,(S)" "(21)/(5)),(,(T)" "2sqrt(6)),(,(U)" "(10)/(3)):}`
Q. Which of the following is the only incorrect combination?

Options

A. (IV)-(U)
B. (III)-(R)
C. (IV)-(S)
D. (I)-(P)

Answer

C

Explanation

(I). `2r=MN=3+sqrt(3^(2)+4^(2))+4=12impliesr=6`
centre C of circle `C_(3)` lies on `y=(4)/(3)x`
Let `C(h,(4)/(3)h)`
`OC=MC-OM=(12)/(2)-3=3`
`thereforesqrt(h^(2)+(16)/(9)h^(2))=3implies(5h)/(3)=3impliesh=(9)/(5)` ltbr. `k=(4)/(3)h=(12)/(5)`
`therefore2h+k=(18)/(5)+(12)/(5)=6`
(II). Equation of line ZW
`C_(1)impliesC_(2)`
`implies3x+4y=9`
Distance of ZW from (0,0)
`(|-9|)/(sqrt(3^(2)+4^(2)))=(9)/(5)`
Length of `XY=2sqrt(3^(2)-((9)/(5))^(2))=(24)/(5)`
Distance of ZW from C
`(|(3xx9)/(5)+4xx(12)/(5)-9|)/(sqrt(3^(2)+4^(2)))=(6)/(5)`
Length of ZE`=2sqrt(6^(2)-(6^(2))/(5^(2)))=(24sqrt(6))/(5)`
`therefore("length of ZW")/("length of XY")=sqrt(6)`
(III). Area of `DeltaMZN=(1)/(2).NM((1)/(2)ZW)=(72sqrt(6))/(5)`
Area of `DeltaZMW=(1)/(2).ZW(OM+OP)=(1)/(2)(24sqrt(6))/(5).(3+(9)/(5))=(288sqrt(6))/(25)`
`therefore("Area of "DeltaMZN)/("Area of "DeltaZMW)=(5)/(4)`
(IV). Slop of tangent of `C_(1)` at `M=(-1)/(4//3)=-(3)/(4)`
`therefore` Equation of tangent `y=mx-3sqrt(1+m^(2))`
`y=-(3)/(4)x-3sqrt(1+(9)/(16))`
`y=(-3x)/(4)-(15)/(4)impliesx=-(4y)/(3)-5` ...(i)
tangent to `x^(2)=4(2alpha)y` is
`x=m'y+(2alpha)/(m')` ..(ii)
compare (i) and (ii)
`m'=-(4)/(3)` and `(2alpha)/(m)=-5impliesalpha=(10)/(3)`