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If \[A(0,3),B(-2,0)\]and \[C(6,1)\]be the vertices of a triangle and \[M(b,b+1)\]be a moving
point then
Number of integral value of \[b\]if \[M\]lies inside the \[\Delta ABC\]
(a) 0
(b) 1
(c) 2
(d) 3

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Last updated date: 18th Jul 2024
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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:
seo images


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.