Statements on Lines as Chords and Diameters of Circle

To determine the correctness of STATEMENT-1 and STATEMENT-2, we analyze the given equations of the lines and the circle step-by-step.

1. Circle Properties

The equation of the given circle is:
$C : x^2 + y^2 + 6x - 10y + 30 = 0$

Comparing this with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we get:

• Center, $O = (-g, -f) = (-3, 5)$
• Radius, $R = \sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + 5^2 - 30} = \sqrt{9 + 25 - 30} = \sqrt{4} = 2$

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2. Analysis of STATEMENT-1

The line $L_1$ is given by:
$L_1 : 2x + 3y + p - 3 = 0$

For $L_1$ to be a chord of circle $C$, the perpendicular distance $d_1$ from the center $O(-3, 5)$ to the line $L_1$ must be less than the radius $R$:
$d_1 = \frac{|2(-3) + 3(5) + p - 3|}{\sqrt{2^2 + 3^2}} < 2$
$\implies \frac{|p + 6|}{\sqrt{13}} < 2 \implies |p + 6| < 2\sqrt{13}$
$\implies p \in (-6 - 2\sqrt{13}, -6 + 2\sqrt{13})$

Since $2\sqrt{13} \approx 7.21$, the range of $p$ for $L_1$ to be a chord is approximately $(-13.21, 1.21)$.

For $L_2 : 2x + 3y + p + 3 = 0$ to be a diameter of circle $C$, it must pass through the center $O(-3, 5)$:
$2(-3) + 3(5) + p + 3 = 0 \implies p + 12 = 0 \implies p = -12$

Thus, $L_2$ is a diameter only when $p = -12$. For any other value of $p$ in the range $(-13.21, 1.21)$ (for instance, $p = 0$), $L_1$ is a chord, but $L_2$ is not a diameter.

Therefore, STATEMENT-1 ("If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$") is TRUE.

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3. Analysis of STATEMENT-2

For line $L_1$ to be a diameter of circle $C$, it must pass through the center $O(-3, 5)$:
$2(-3) + 3(5) + p - 3 = 0 \implies p + 6 = 0 \implies p = -6$

Substituting $p = -6$ into the equation for $L_2$:
$L_2 : 2x + 3y + (-6) + 3 = 0 \implies 2x + 3y - 3 = 0$

Now, let's find the perpendicular distance $d_2$ from the center $O(-3, 5)$ to $L_2$:
$d_2 = \frac{|2(-3) + 3(5) - 3|}{\sqrt{2^2 + 3^2}} = \frac{|-6 + 15 - 3|}{\sqrt{13}} = \frac{6}{\sqrt{13}}$

Comparing $d_2$ with the radius $R = 2$:
$d_2^2 = \frac{36}{13} \approx 2.77$
$R^2 = 4$

Since $d_2^2 < R^2 \implies d_2 < R$, the line $L_2$ intersects the circle at two distinct points, meaning $L_2$ is a chord of circle $C$.

Therefore, STATEMENT-2 ("If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$") is FALSE.

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Conclusion

• STATEMENT-1 is True
• STATEMENT-2 is False

Thus, the correct choice is Option C.