Question

If the points A(1,2), O(0,0) and C(a,b) are collinear, then
(a). a=b
(b). a=2b
(c). 2a=b
(d). a=-b

Answer 1

Hint: In this question, three points are given to be collinear, i.e. their coordinates should satisfy the equation of a straight line. Now, if we consider the general equation of a straight line, then the coordinates of all the given points will satisfy its equation and thus we can obtain three linear equations. As, there are two more variables in the general equation of a straight line, we can solve the equations to find the relation between a and b.

Complete step-by-step solution -
In this question, we are given three points which are collinear, i.e. they lie on the same straight line. Now, the general equation of a straight line can be written as
$y=mx+c.................(1.1)$
where m is the slope and c is the intercept of the straight line.
If the points A(1,2), O(0,0) and C(a,b) lie on a straight line of the form given in equation (1.1), their coordinates should satisfy equation (1.1), thus, we obtain the equations
$2=m\times 1+c\Rightarrow m=2-c............(1.2)$
And
$0=m\times 0+c\Rightarrow c=0.............(1.3)$
Using the value of c in equation (1.2), we get
$m=2-0=2............(1.4)$
Thus, using the values of equations (1.3) and (1.4) in equation (1.1), the equation of the straight line in which the points A(1,2), O(0,0) lie is given by
$y=2x+0\Rightarrow y=2x.............(1.5)$
Now, as C(a,b) is collinear with A(1,2), O(0,0), it should also satisfy the equation of the straight line given in equation(1.5). Thus, we obtain the relation
$b=2a$
which matches option (c) in the question. Thus (b) is the correct answer.

Note: In equations (1.2) and (1.3), we should consider the points A(1,2), O(0,0) to find m and c and not the point C(a,b) as its coordinates are itself variables and thus, would not help us to find fixed values of m and c.