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The value of \[m\] for which the lines \[3x = y - 8\] and \[6x + my + 16 = 0\] coincide is
A.2
B.\[ - 2\]
C.\[\dfrac{1}{2}\]
D.\[ - \dfrac{1}{2}\]

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Last updated date: 22nd Jun 2024
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Answer
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Hint: First we will find the slope of the tangent is the differentiation of the given curves with respect to \[x\] and then we will use that for the given equations to coincide, the slope should be equal.

Complete step-by-step answer:
We are given that the lines \[3x = y - 8\] and \[6x + my + 16 = 0\] coincide.
Rewriting the given equations, we get
\[y = 3x + 8{\text{ ......eq.(1)}}\]
\[y = - \dfrac{6}{m}x - \dfrac{{16}}{m}{\text{ ......eq(2)}}\]
We know that the slope of the tangent is the differentiation of the given curve with respect to \[x\].
Differentiating the equation (1) with respect to \[x\], we get
\[
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {3x + 8} \right) \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}3x - \dfrac{d}{{dx}}8 \\
   \Rightarrow \dfrac{{dy}}{{dx}} = 3{\text{ ......eq.(3)}} \\
 \]
Differentiating the equation (2) with respect to \[x\], we get
\[
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( { - \dfrac{6}{m}x - \dfrac{{16}}{m}} \right) \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( { - \dfrac{6}{m}x} \right) - \dfrac{d}{{dx}}\left( { - \dfrac{{16}}{m}} \right) \\
   \Rightarrow \dfrac{{dy}}{{dx}} = - \dfrac{6}{m}{\text{ ......eq.(4)}} \\
 \]
For the given equations to coincide, the slope should be equal.
So, taking equation (3) and equation (4) equal, we get
\[ \Rightarrow 3 = - \dfrac{6}{m}\]
Cross-multiplying the above equation, we get
\[ \Rightarrow 3m = - 6\]
Dividing the above equation by 3 on both sides, we get
\[ \Rightarrow m = - 2\]
Hence, option B is correct.

Note: The slope equals the rise divided by the run. You can determine the slope of a line from its graph by looking at the rise and run. One characteristic of a line is that its slope is constant all the way along it. So, you can choose any 2 points along the graph of the line to figure out the slope. Avoid calculation mistakes.