Assertion :Each point on the line $y - x + 12 = 0$ is equidistant from the lines $4 y + 3 x - 12 = 0$ , $3 y + 4 x - 24 = 0$ Reason: The locus of a point which is equidistant from two given lines is the angular bisector of the two lines.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
The straight line equidistant from a pair of straight lines is the angle bisector of the angle between the straight lines.
Hence, consider the equations $3 x + 4 y - 12 = 0$ and $4 x + 3 y - 24 = 0$
Equation of the angle bisector is given by
$| \frac{3 x + 4 y - 12}{\sqrt{3^{2} + 4^{2}}} | = | \frac{3 y + 4 x - 24}{\sqrt{3^{2} + 4^{2}}} |$
$| \frac{3 x + 4 y - 12}{5} | = | \frac{3 y + 4 x - 24}{5} |$
$| 3 x + 4 y - 12 | = | 4 x + 3 y - 24 |$
$3 x + 4 y - 12 = 4 x + 3 y - 24$ ....(i)
$x - y = 12$ is the equation of one of the angle bisectors.
$3 x + 4 y - 12 = - 4 x - 3 y + 24$
$7 x + 7 y = 12$ ...(ii) is the equation of the second angle bisector.
both assertion and reasons are true, and the reason is the correct explanation of the assertion.

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