Answer

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Hint: Find the extremes of the given point by using the vertices of the triangle.

The figure for the given problem is as follows:

From the above figure it is clear that the x-coordinate of point M should lie between \[(-2,6)\], as they

are coordinates of points B and C the extremes.

Therefore,

\[-2<b<6.......(i)\]

Similarly, the y-coordinates of point M should lie between \[(0,3)\], as they are the coordinates of the

points A and B the extremes.

Therefore,

\[0<b+1<3\]

Subtracting ‘1’ from above, we get

\[0-1<b+1-1<3-1\]

\[\Rightarrow -1<b<2\]

This equation satisfies the equation (i). So, the possible values of ‘b’ are \[(0,1)\].

So, the number of integral values of \[b\]for \[M\]to lie inside the \[\Delta ABC\]is \[2\].

And the point \[M\] can be \[(0,1)\] and \[(1,2)\] .

Hence the correct answer is option (c).

Note: We can solve this by finding the equations of all the three sides then applying the condition for

two points lying on the same side. This will be a lengthy process.

The figure for the given problem is as follows:

From the above figure it is clear that the x-coordinate of point M should lie between \[(-2,6)\], as they

are coordinates of points B and C the extremes.

Therefore,

\[-2<b<6.......(i)\]

Similarly, the y-coordinates of point M should lie between \[(0,3)\], as they are the coordinates of the

points A and B the extremes.

Therefore,

\[0<b+1<3\]

Subtracting ‘1’ from above, we get

\[0-1<b+1-1<3-1\]

\[\Rightarrow -1<b<2\]

This equation satisfies the equation (i). So, the possible values of ‘b’ are \[(0,1)\].

So, the number of integral values of \[b\]for \[M\]to lie inside the \[\Delta ABC\]is \[2\].

And the point \[M\] can be \[(0,1)\] and \[(1,2)\] .

Hence the correct answer is option (c).

Note: We can solve this by finding the equations of all the three sides then applying the condition for

two points lying on the same side. This will be a lengthy process.

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