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?

 Apr 30, 2019

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



cool cool cool

 May 1, 2019

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