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

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Statement 1 is true and statement 2 is false.

Statement 1 is false and statement 2 is true.

Statement 1 and statement 2 is true.

Statement 1 and statement 2 is false.

answer is A.

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Detailed Solution

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

So, the radius = 2

Distance between the parallel lines $L_{1} and L_{2}$ is,

$|\frac{(p + 3) - (p - 3)}{\sqrt{4 + 9}}| = \frac{6}{\sqrt{13}} < radius (2)$

So, the Statement II is false, but the Statement I is true.

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