If the points A(x, 2), B (-3,-4) and C (7-5) are collinear, then the value of x is:
$( A) \ -63$
$( B) \ 63$
$( C) \ 60$
$( D) \ -60$
Given: Points $A( x,\ 2) ,\ B\ ( -3,-4)$ and $C( 7-5)$ are collinear.
To do: To find the value of x.
Solution:
It is given that the three points $\displaystyle A( x,\ 2) ,\ B( -3,-4) \ and\ ( 7,-5)$ are collinear.
If given points are collinear, then area formed by $A( x,\ 2) ,\ B( -3,-4)$ and $( 7,-5)$ will be 0.
And we know that area of a triangle with vertices $( x_{1} ,\ y_{1}) ,\ ( x_{2} ,\ y)$ and $( x_{3} ,\ y_{3} )$
$\frac{1}{2}[ x_{1}( y_{2} -y_{3}) +x_{2}( y_{3} -y_{1}) +x_{3}( y_{1} -y_{2})]$
Here on subtituting the values in the formula,
$\frac{1}{2}[ x( -4+5) -3( -5-2) +7( 2+4) =0$
$\frac{1}{2}( x+21+42) =0$
$x+63=0$
$x=-63$
Thus, the value of x is - 63.
Hence, The correct option is $( A)$.
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