Consider $L_1 : 2x + 3y + p - 3 = 0$ $L_2 : 2x + 3y + p + 3 = 0$ where $p$ is a real number, and $C : x^{2} + y^{2} + 6x - 10y + 30 = 0$ STATEMENT - 1: If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$ and STATEMENT - 2: If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$

Consider L₁ : 2x + 3y + p - 3 = 0 L₂ : 2x + 3y + p + 3 = 0 where p is a real number, and C : x² + y² + 6x - 10y + 30 = 0 STATEMENT - 1: If line L₁ is a chord of circle C, then line L₂ is not always a diameter of circle C and STATEMENT - 2: If line L₁ is a diameter of circle C, then line L₂ is not a chord of circle C

$C : x^{2} + y^{2} + 6x - 10y + 30 = 0$

$(x+3)^{2} + (y-5)^{2} = 4 \Rightarrow \text{centre } (-3, 5),\ r = 2$

$L_1 : 2x + 3y = 3 - p,\qquad L_2 : 2x + 3y = -3 - p\ \text{(parallel)}$

$\text{Decide: truth of Statement-1 and Statement-2,}$

$\text{and whether Statement-2 explains Statement-1}$

$C: x^{2}+y^{2}+6x-10y+30=0$

$(x+3)^{2}+(y-5)^{2}=4$

$\text{Centre } O(-3,5),\quad r=2$

$L_1: 2x+3y = 3-p,\qquad L_2: 2x+3y = -3-p$

$\text{Distance between } L_1, L_2 = \dfrac{6}{\sqrt{13}} \approx 1.66 < 2$

$\text{1. Distances } d_1, d_2 \text{ of } L_1, L_2 \text{ from } O$

$\text{2. Chord } \iff d<2;\quad \text{Diameter} \iff d=0$

$\text{3. Test Statement-1, then Statement-2}$

$d = \dfrac{|a x_0 + b y_0 + c|}{\sqrt{a^{2}+b^{2}}}$

$d_1 = \dfrac{|p+6|}{\sqrt{13}},\qquad d_2 = \dfrac{|p+12|}{\sqrt{13}}$

$\text{Chord} \iff d < 2$

$\text{Diameter} \iff d = 0$

$p=-12:\ d_2 = 0 \Rightarrow L_2 \text{ is a diameter}$

$d_1 = \dfrac{6}{\sqrt{13}} < 2 \Rightarrow L_1 \text{ is a chord}$

$\therefore\ \text{Statement-1 is TRUE}$

$p=-6:\ d_1 = 0 \Rightarrow L_1 \text{ is a diameter}$

$d_2 = \dfrac{6}{\sqrt{13}} < 2 \Rightarrow L_2 \text{ is a chord}$

$\therefore\ \text{Statement-2 is FALSE}$

$\text{Statement-1: TRUE},\qquad \text{Statement-2: FALSE}$

$\text{Answer: S-1 True,\ S-2 False}$