Angle ate the area

Geometry Level 2

There are two circles. One with centre A [say circle A] whose radius is fixed and cannot be changed and one with center C [say circle C] whose radius can be varied with the help of a point B which lies on the circumference of the circle with center A.

Right now point A, B and C are collinear as shown(assume 180°)

Now the point B is moved along the circumference of Circle A in anticlockwise direction such that angle C A B = 9 0 ∘ CAB=90^\circ (assume) as shown in figure.

What is the ratio of the area of Circle C before movement of point B to area of Circle C after the movement of the point B

NOTE: The point B is bound to move only on the circumference of the circle A. And the point C stays at one point and cannot move.

These figures are just to give an idea about the question, so the assumptions can be taken as answers will be with respect to the assumptions.

1:4 4:1 2:1 1:2

Abhay Tiwari by Abhay Tiwari , 28, India

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1 solution

Ashish Menon
Apr 20, 2016

Let the diameter of circle with center A be 'r' cm, then the area of the circle with center C is π × r 2 \pi × r^2

Now, in second case, the radous of circle with center C would be r 2 4 + r 2 = 2 r \sqrt{\dfrac{r^2}{4} + \dfrac{r}{\sqrt{2}}} = \sqrt{2}r
So, the area of circle with center C = π × r 2 2 \pi × {\dfrac{r}{\sqrt{2}}}^2 = π × r 2 2 \pi × \dfrac{r^2}{2}

So, the ratio of areas = π × r 2 π × r 2 2 = 2 : 1 \dfrac{\pi × r^2}{\dfrac{\pi × r^2}{2}} = \boxed{2:1}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You are absolutely right.!

Abhay Tiwari - 5 years, 1 month ago

Here you go

Abhay Tiwari - 5 years, 1 month ago

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Seems like a great question :) But, I didnt understand what to find? :+1:

Ashish Menon - 5 years, 1 month ago

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You have to find the angle subtended by the arc traversed by point B at the center of Circle A as it moves from initial to final position.

Abhay Tiwari - 5 years, 1 month ago

Ashish I have made some correction, you can check it now

Abhay Tiwari - 5 years, 1 month ago

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@Abhay Tiwari – XD, I understood it when you told it, I am still solving your problem to get the hang of it. But now maybe after lunch. ;)

Ashish Menon - 5 years, 1 month ago

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