azureus88 said:
The ratio of the length of a chord of a circle to the diameter is x:1. The chord moves around the circle so that its length is unchanged (forming the locus of another circle). find the rato of hte areas of the 2 circles.
my answer to the question is (1 - x^2): 1
the real answer is 0.25(4 - x^2):1
so how do you get the answer ?
If the length of the chord is x, then the diameter of the circle is 1, thus its radius is 0.5, so its area is π/4 square units.
So first consider a circle of radius 0.5 units. Now the locus created by the chord is a circle with the radius equal to the perpendicular distance from the centre of this circle to the chord. Let this radius be r.
Since the line drawn from the centre to the perpendicular of the chord bisects the chord, a right angled triangle can be formed with sides r, 0.5 and x/2.
=> 0.25 = r² + x²/4 by Pythagoras' theorem
.: r² = (1 - x²) / 4
So the area of a circle with radius r is:
πr² = π(1 - x²) / 4
Ratio of the original circle to the locus is:
π(1 - x²) / 4 : π/4
=> (1 - x²) : 1
So I think you're right. You get the other answer if the ratio of the chord to the RADIUS is x:1 rather than the diameter...
