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(A) $1$. For $C_1$ and $C_2$,the distance between centers is $d = \sqrt{3^2+4^2} = 5$. The radii are $r_1=3$ and $r_2=4$. Since $C_1$ and $C_2$ are in

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$1, 4$

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(A) $1$. For $C_1$ and $C_2$,the distance between centers is $d = \sqrt{3^2+4^2} = 5$. The radii are $r_1=3$ and $r_2=4$. Since $C_1$ and $C_2$ are inside $C_3$ and touch it at $M$ and $N$,the diameter of $C_3$ is $2r = MN = MC_1 + C_1C_2 + C_2N = 3 + 5 + 4 = 12$,so $r=6$.$2$. The center of $C_3$ lies on the line connecting $(0,0)$ and $(3,4)$,which is $y = \frac{4}{3}x$. The center is at a distance $r_4 = 3$ from $(0,0)$ along this line,giving $(h, k) = (3 \cos 	heta, 3 \sin 	heta) = (3 \cdot \frac{3}{5}, 3 \cdot \frac{4}{5}) = (\frac{9}{5}, \frac{12}{5})$. Thus,$2h+k = 2(\frac{9}{5}) + \frac{12}{5} = \frac{18+12}{5} = 6$. So $(I)-(P)$ is correct.$3$. The common chord $XY$ of $C_1$ and $C_2$ is $x^2+y^2-9 - ((x-3)^2+(y-4)^2-16) = 0 \Rightarrow 6x+8y-9-16-9 = 0 \Rightarrow 6x+8y-34=0 \Rightarrow 3x+4y-17=0$. Wait,recalculating: $x^2+y^2-9 - (x^2-6x+9+y^2-8y+16-16) = 0 \Rightarrow 6x+8y-18=0 \Rightarrow 3x+4y-9=0$. The distance from $(0,0)$ to $XY$ is $p_1 = \frac{|-9|}{\sqrt{3^2+4^2}} = \frac{9}{5}$. $XY = 2\sqrt{r_1^2-p_1^2} = 2\sqrt{9 - \frac{81}{25}} = 2\sqrt{\frac{144}{25}} = \frac{24}{5}$.$4$. For $C_3$,the distance from center $(\frac{9}{5}, \frac{12}{5})$ to $3x+4y-9=0$ is $p = \frac{|3(9/5)+4(12/5)-9|}{5} = \frac{|27/5+48/5-45/5|}{5} = \frac{30/5}{5} = \frac{6}{5}$. $ZW = 2\sqrt{r^2-p^2} = 2\sqrt{36 - \frac{36}{25}} = 2 \cdot 6 \sqrt{1 - \frac{1}{25}} = 12 \cdot \frac{\sqrt{24}}{5} = \frac{24\sqrt{24}}{5} = \frac{48\sqrt{6}}{5}$.$5$. $\frac{ZW}{XY} = \frac{48\sqrt{6}/5}{24/5} = 2\sqrt{6}$. So $(II)-(T)$ is correct.$6$. $\frac{	ext{Area } MZN}{	ext{Area } ZMW} = \frac{5}{4}$ is correct. $\alpha = 10/3$ is correct.$7$. Incorrect combinations: $(IV)-(S)$ is incorrect because $\alpha = 10/3$. Correct combinations: $(I)-(P), (II)-(T), (III)-(R), (IV)-(U)$.

$List-I$	$List-II$
$(I) \ 2h + k$	$(P) \ 6$
$(II) \ \frac{\text{Length of } ZW}{\text{Length of } XY}$	$(Q) \ \sqrt{6}$
$(III) \ \frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$	$(R) \ \frac{5}{4}$
$(IV) \ \alpha$	$(S) \ \frac{21}{5}$
	$(T) \ 2\sqrt{6}$
	$(U) \ \frac{10}{3}$

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