Determine algebraically if the perpendicular bisector of chord AB passes through the point (6,1).Please help - I am unsure of how to do this! Thanks :)

Expert Answers

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I assume that you know, or can find, the center of the circle and that you know the coordinates for A and B.

 

First, the perpendicular bisector of `bar(AB)` must go through the midpoint of `bar(AB)` . If A is at `(x_1,y_1)` and B is at `(x_2,y_2)` , then the midpoint is at the point `((x_1+x_2)/2,(y_1+y_2)/2)` .

Next, the diameter of the circle lies on the perpendicular bisector of the chord.(From geometry we know that a diameter perpendicular to a chord bisects a chord, and the converse). Thus the center of the circle lies on the perpendicular bisector. Let the center be at `(x_0,y_0)` .

Now you have two points on the perpendicular bisector; if either of these points is (6,1) you are done. If not, find the equation of the line through `((x_1+x_2)/2,(y_1+y_2)/2)` and `(x_0,y_0)` . From this equation you can determine if (6,1) is on the line.

 

Hope this helps.

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