The circle described has a radius of 1, and is centred on the origin (0,0).g4gmat wrote:What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = 3/4*x - 3?
A) 1.4
B) sqrt (2)
C) 1.7
D) sqrt (3)
E) 2.0
answer a
The line described intercepts the y-axis at (0,-3), and the x-axis at (4,0). Therefore the line, plus the x- and y-axes define a right-angled triangle with sides of length 3, 4, and 5.
The height of the triangle (with respect to the long edge) is a line segment from the origin to the line y = 3/4*x - 3. Part of that height is within the radius of the circle, and the rest is the minimum distance from the circle to the line.
The part within the radius must be 1 unit long. The easiest way to calculate the height of the triangle that I can see is to use the area formula, area = 0.5 * b * h
Firstly area = 0.5 * 3 * 4 = 6
And then, keeping area constant, but using the long edge as the base:
area = 0.5 * b * h
6 = 0.5 * 5 * h
h = 12 / 5 = 2.4
Remember to subtract the 1 that is within the radius of the circle, giving an answer of 1.4
