SOLUTION: Two circles, one of radius 1 and the other of radius 2, touch externally at P.
A straight line through P cuts the area formed by these two circles in the ratio 1:2.
In what ratio
Algebra ->
Rational-functions
-> SOLUTION: Two circles, one of radius 1 and the other of radius 2, touch externally at P.
A straight line through P cuts the area formed by these two circles in the ratio 1:2.
In what ratio
Log On
Question 1182794: Two circles, one of radius 1 and the other of radius 2, touch externally at P.
A straight line through P cuts the area formed by these two circles in the ratio 1:2.
In what ratio does this line cut the area of the smaller circle? Answer by greenestamps(13200) (Show Source):
Here is a sketch of the two circles and the line through their point of tangency:
The figures of the small circle with its chord and of the large circle and its chord are similar figures.
The radius of the small circle is 1; the radius of the large circle is 2. So the areas of the two circles are pi and 4pi.
Let x be the area of the smaller portion of the small circle; then the area of the larger portion is (pi-x).
Since the scale factor between the two circles is 1:2, the ratio of corresponding area measurements between the two circles is 1:4. So the smaller portion of the large circle is 4x, and the larger portion of the large circle is 4(pi-x) = (4pi-4x).
We are told that the line divides the total area of the two circles in the ratio 1:2. The total area is 5pi. That means the area of the smaller portion of the small circle, plus the larger portion of the area of the large circle, is 2/3 of the total 5pi:
The area of the smaller portion of the small circle is x=(2/9)pi; that means the area of the larger portion of the small circle is (pi-x) = (7/9)pi.
And that means the smaller circle is divided in the ratio (2/9):(7/9) = 2:7
