A) \[\frac{{{c}_{1}}-{{c}_{2}}}{\sqrt{{{m}^{2}}+1}}\]
B) \[\frac{{{c}_{2}}-{{c}_{1}}}{\sqrt{{{m}^{2}}+1}}\]
C) \[\frac{{{c}_{2}}\tilde{\ }{{c}_{1}}}{\sqrt{{{m}^{2}}+1}}\]
D) 0
Correct Answer: C
Solution :
Since, lines are parallel, the distance from origin to the lines are \[{{d}_{1}}=\frac{{{c}_{1}}}{\sqrt{1+{{m}^{2}}}},{{d}_{2}}=\frac{{{c}_{2}}}{\sqrt{1+{{m}^{2}}}}\] \[\therefore \]Perpendicular distance between lines is \[{{d}_{1}}\tilde{\ }{{d}_{2}}=\frac{{{c}_{1}}\tilde{\ }{{c}_{2}}}{\sqrt{1+{{m}^{2}}}}\] Alternate Solution: Let us take the assuming point on the first line is \[(0,{{c}_{1}}).\] The perpendicular distance from point \[(0,{{c}_{1}})\] to the line \[y=m\text{ }x+{{c}_{2}}\] \[=\frac{{{c}_{1}}\tilde{\ }{{c}_{2}}}{\sqrt{1+{{m}^{2}}}}\]
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