Statement 1: If the point $(2 a - 5, a^{2})$ is on the same side of the line $x + y - 3 = 0$ as that of the origin, then $a \in (2, 4) .$

Statement 2: The point $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ lie on the same or opposite sides of the line $a x + b y + c = 0,$ as $a x_{1} + b y_{1} + c$ and $a x_{2} + b y_{2} + c$ have the same or opposite signs.

Statement 1 is False and Statement 2 is True

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Both the statements are True but Statement 2 is not the cirrect explanation of Statement 1.

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Statement 1 is True and Statement 2 is False

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Both the statements are True but Statement 2 is the cirrect explanation of Statement 1.

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Solution

The correct option is A Statement 1 is False and Statement 2 is True
Let $L : x + y - 3 = 0$
$L (0, 0) = - 3$ and
$L (2 a - 5, a^{2}) = 2 a - 5 + a^{2} - 3$
$∵ (0, 0) and (2 a - 5, a^{2})$ lie on the same side of the given line, so
$L (0, 0) \times L (2 a - 5, a^{2}) > 0 \Rightarrow - 3 (a^{2} + 2 a - 8) > 0 \Rightarrow (a + 4) (a - 2) < 0 \Rightarrow a \in (- 4, 2)$

Second statement is correct
Hence, option $(D)$ is the correct option.

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