Point P splits a diameter of a circle into segments of lengths 2 and 6. What is the shortest distance from the center of this circle to a chord through P that makes a 30-degree angle with the diameter?
See the following, SydSu.....
A perpendicular bisector drawn to the chord from the center of the circle will be the shortest distance from the center to the chord.....
Let the equation of the circle be x^2 + y^2 = 16
The equation of the line containing the chord is y = (1/sqrt(3)) ( x + 2)
Call the point of intersection of the bisector and the chord, M
Then....AMP will form a right triangle.....with angle AMP = 90° and AP the hypotenuse = 2
Then...AM will be the distance we are looking for.....and since this side of the right triangle is opposite the 30° it will be (1/2) of AP = 1