Question

$\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Centre(s) of the circle(s) having radius}5\text{and}& \text{(P)}& (4,6)\\ & \text{touching the line}3x+4y-11=0\text{at}(1,2)\text{is (are)}& & \\ \text{(B)}& \text{End points of one of the diameters of}& \text{(Q)}& (1,1)\\ & x^2+y^2-6x-8y+20=0\text{are}& & \\ \text{(C)}& \text{The line equidistant from both the lines}& \text{(R)}& (3,-3)\\ & 4x+2y+2=0\text{and}6x+3y-21=0,& & \\ & \text{passes through}& & \\ \text{(D)}& \text{If one of the sides of a square is}3x-4y-12=0& \text{(S)}& (-2,7)\\ & \text{and its centre is}(0,0),\text{then its diagonal}& & \\ & \text{passes through}& & \\ & & \text{(T)}& (-2,-2)\end{array}$

Which of the following is the only CORRECT combination?

• A. $\text{(A)}\to \text{(P)},\text{(S)}$
• B. $\text{(A)}\to \text{(P)},\text{(T)}$
• C. $\text{(B)}\to \text{(P)},\text{(R)}$
• D. $\text{(B)}\to \text{(Q)},\text{(T)}$

Answer 1

The correct option is B $\text{(A)}\to \text{(P)},\text{(T)}$

$\left(A\right)$

Slope of tangent $=-\frac{3}{4}$
$\Rightarrow$ Slope of $C_1{C}_2=\frac{4}{3}=\tan \theta$
$\Rightarrow \cos \theta =\frac{3}{5},\sin \theta =\frac{4}{5}$
$\therefore$ By parametric form,
$C_1,C_2\equiv (1\pm 5\cos \theta ,2\pm 5\sin \theta )$
$=(1\pm 3,2\pm 4)=(4,6),(-2,-2)$

$\left(B\right)$
Given circle is $x^2+y^2-6x-8y+20=0$
$\therefore$ Centre $\equiv (3,4),$ which does not coincide with the mid-point of $PR$ (or) $QT$