NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 7: Coordinate Geometry - Exercise 7.1
Question 20

Question. 20

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)

\(a=-b\)

Step 1: Three points are collinear if they lie on the same straight line. This means the slope between any two pairs of points must be equal.

Step 2: First, find the slope of the line joining \(O(0,0)\) and \(A(1,2)\):

\[ m_{OA} = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{2 - 0}{1 - 0} = \dfrac{2}{1} = 2 \]

Step 3: Next, find the slope of the line joining \(O(0,0)\) and \(C(a,b)\):

\[ m_{OC} = \dfrac{b - 0}{a - 0} = \dfrac{b}{a} \]

Step 4: Since the points are collinear, both slopes must be equal:

\[ \dfrac{b}{a} = 2 \]

Step 5: Multiply both sides by \(a\):

\[ b = 2a \]

Step 6: Rearranging gives:

\[ 2a = b \]

Final Answer: Option C (\(2a = b\)).