Question

Let $S_1$ and $S_2$ be circles of radii 1 and r (r > 1) respectively touching the coordinate axes.
Column-1: Conditions between circles $S_1$ and $S_2$
Column-2: Values of r for conditions in Column-1.
Column-3: Number of common tangents between S1 and S2 for conditions in column-1.
$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \text{(I)}& S_2\text{passes through the centre}& \text{(i)}& 3& \text{(P)}& 1\\ & \text{of}{S}_1.& & & & \\ \text{(II)}& S_1\text{and}{S}_2\text{touch each other}& \text{(ii)}& 2+\sqrt{2}& \text{(Q)}& 2\\ \text{(III)}& S_1\text{and}{S}_2\text{are orthogonal}& \text{(iii)}& 2+\sqrt{3}& \text{(R)}& 3\\ \text{(IV)}& S_1\text{and}{S}_2\text{have longest}& \text{(iv)}& 3+2\sqrt{2}& \text{(S)}& 4\\ & \text{common chord}& & & & \end{array}$

Which of the following options is the only CORRECT combination?

• A. (I) (ii) (Q)
• B. (I) (ii) (P)
• C. (I) (ii) (R)
• D. (II) (ii) (S)

Answer 1

The correct option is A

(I) (ii) (Q)

$S_1:(x-1{)}^2+(y-1{)}^2=1^2$

$\Rightarrow x^2+y^2-2x-2y+1=0$

Centre $=(1,1)=C_1,r_1=1$

$S_2:(x-r{)}^2+(y-r{)}^2=r^2\Rightarrow x^2+y^2-2rx-2ry+r^2=0$

Centre $=(r,r)=C_2,r_2=r$

(A) $S_2=0$ passes through (1, 1)

$r=2+\sqrt{2}$

Number of common tangents = 2

(B) $C_1{C}_2=r_1+r_2$

$\Rightarrow r=3+2\sqrt{2}$

Number of common tangents = 3

(C) For orthogonal,

$2g_1{g}_2+2f_1{f}_2=c_1+c_2$

$\Rightarrow 2r+2r=r^2+1$

$\Rightarrow r=2\pm \sqrt{3}$

$\therefore r=2\pm \sqrt{3}$ ( r > 1)

Number of common tangents = 2

(D) For largest common chord, common chord between $S_1=0$ and $S_2=0$ will become diameter of $S_1=0$.
Equation of common chord is $S_1-S_2=0$ and passes through (1, 1). So, r = 3.
Number of common tangents = 2