The points \((0,5),\ (0,-9),\ (3,6)\) are collinear.
False.
Step 1: Three points are collinear if the area of the triangle formed by them is 0.
Step 2: Use the area of triangle formula for points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\):
\(\text{Area} = \dfrac{1}{2}\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right|\).
Step 3: Let
Step 4: Substitute into the expression inside absolute value:
\(x_1(y_2-y_3) = 0\big((-9)-6\big) = 0\)
\(x_2(y_3-y_1) = 0\big(6-5\big) = 0\)
\(x_3(y_1-y_2) = 3\big(5-(-9)\big) = 3\times 14 = 42\)
Step 5: Add them:
\(0 + 0 + 42 = 42\)
Step 6: Area of the triangle:
\(\text{Area} = \dfrac{1}{2}|42| = 21\)
Final Step: Since the area is \(21\) (not zero), the points do not lie on the same straight line. Therefore, the statement is False.