Question

Statement-1: Each point on the line y-x+12=0 is at the same distance from the lines 3x+4y-12=0 and 4x+3y-12=0.
Statement-2: Locus of point which is at an equal distance from two intersecting given lines is the angle bisector of the two lines.

(a) Statement 1 is true; Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
(b) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.
(c) Statement 1 is true; Statement 2 is false.
(d) Statement 1 is false; Statement 2 is true.

Answer 1

Hint: The angle bisector of two lines is the line which is at an equal distance from both the lines. We need to substitute the equation of the lines in the formula. There will be two cases for the plus and minus sign each. Hence, there are two equations of angle bisectors, which are perpendicular to each other which are given as-
$\dfrac{{\mathrm a}_1\mathrm x+{\mathrm b}_1\mathrm y+{\mathrm c}_1}{\sqrt{{\mathrm a}_1^2+{\mathrm b}_1^2}}=\pm\left(\dfrac{{\mathrm a}_2\mathrm x+{\mathrm b}_2\mathrm y+{\mathrm c}_2}{\sqrt{{\mathrm a}_2^2+{\mathrm b}_2^2}}\right)\;\;\;​\;\;$

Complete step-by-step answer:
From the definition of angle bisector, it is clearly visible that Statement-2 is correct.

Now, we will find the angle bisector of the two given lines using its formula.
$\dfrac{3\mathrm x+4\mathrm y-12}{\sqrt{3^2+4^2}}=\dfrac{4\mathrm x+3\mathrm y-12}{\sqrt{4^2+3^2}}\\\mathrm x-\mathrm y=0\;\mathrm{or},\\\dfrac{3\mathrm x+4\mathrm y-12}{\sqrt{3^2+4^2}}=\dfrac{-4\mathrm x-3\mathrm y+12}{\sqrt{4^2+3^2}}\\7\mathrm x+7\mathrm y-24=0$
Hence, both the equations are not satisfied. Statement-1 is false.

Hence, the correct option is (d) Statement 1 is false; Statement 2 is true.

Note: In such type of statement-based problems, first try to find the statement which might be false. If we find the false statement, then we can mark the correct answer without solving the other statement. If both are true, then check if Statement-2 is an explanation of Statement-1 or not.