Question

Find the value of x for which the points ( x , –1), (2, 1) and (4, 5) are collinear.

Answer 1

It is given that the points ( x,−1 ), ( 2,1 ), and ( 4,5 ) are collinear to each other.

Let, P, Q and R be the three collinear points as ( x,−1 ), ( 2,1 ), and ( 4,5 ) respectively.

Now the formula of slope of a line passing through two different points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (1)

Let m PQ and m QR be the slope of the line segment PQ and QR.

Substitute the values ( x,−1 ), ( 2,1 ) for ( x 1 , y 1 ) and ( x 2 , y 2 ) in equation (1) to obtain the slope of line segment PQ.

m PQ = 1−( −1 ) 2−x = 1+1 2−x = 2 2−x

Similarly substitute the values ( 2,1 ), ( 4,5 ) for ( x 1 , y 1 ) and ( x 2 , y 2 ) in equation (1) to obtain the slope of line segment QR.

m QR = 5−1 4−2 = 4 2 =2

The condition for colinearity of three points is,

m PQ = m QR 2 2−x =2 2=2⋅( 2−x ) 2=4−2x

Further simplify,

2x=4−2 2x=2 x=1

Thus, the required value of x for the three points ( x,−1 ), ( 2,1 ), and ( 4,5 ) to be collinear is 1.