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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
 #1
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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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