Question

wrong answers will be reported​

Answer 1

Answer:

Option (i) (a(1-2⁻ⁿ) , b(1-2⁻ⁿ))

Step-by-step explanation:

Given: The co-ordinates of point A are (a,b) and O is the origin. Also, mid-point of OA is A₁, mid-point of AA₁ is A₂ and so on upto Aₙ.

w.k.t., mid-point of line joining two-point P(x₁,y₁) and Q(x₂,y₂) is $\sf{ \dfrac{x_{1}+x_{2}}{2} \ ,\ \dfrac{y_{1}+y_{2}}{2} }$

So, co-ordinates of point A₁ = $\sf{( \dfrac{a+0}{2} \ ,\ \dfrac{b+0}{2} ) = (\dfrac{a}{2},\dfrac{b}{2})}$

Similarly, co-ordinates of point A₂ = $\sf{( \dfrac{\dfrac{a}{2}+a}{2} \ ,\ \dfrac{\dfrac{b}{2}+b}{2} ) = (\dfrac{3a}{4},\dfrac{3b}{4}) }$

And, co-ordinates of point A₃ = $\sf{( \dfrac{\dfrac{3a}{4}+a}{2} \ ,\ \dfrac{\dfrac{3b}{4}+b}{2} ) = (\dfrac{7a}{8},\dfrac{7b}{8}) }$

Also, co-ordinates of point A₄ = $\sf{( \dfrac{\dfrac{7a}{8}+a}{2} \ ,\ \dfrac{\dfrac{7b}{8}+b}{2} ) = (\dfrac{15a}{16},\dfrac{15b}{16}) }$

By observing the above four points and the given options, we can write the general co-ordinates of mid point.

So, the co-ordinates of point Aₙ are (a(1-2⁻ⁿ) , b(1-2⁻ⁿ)).