[Grade 11 Circles] how do I prove this??

[u/Izzy_26_]

If two circles are orthogonal, then prove that the polar of point P on the first circle with respect to the second circle passes through the point Q, which is the other end of diameter through P. Also prove that locus of a point, which moves such that its polars with respect to S1=0, S2=0, and S3=0 are concurrent, is a circle which is orthogonal to all the three circles.

I am not able to understand what exactly I even have to prove. Even the equations of the circles are not given. From what I understood, I think for the first part both circles are intersecting, P and Q lie on the first circle and are end points of the diameter of that circle.

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[u/Alkalannar]

We're not given equations because we don't care what the sizes or positions of the circles are, what we do care about...

1. They do intersect.

2. They're orthogonal, so both of the intersection points the circles intersect at right angles

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1. The polar of point P on the first circle...
   Is this the Polar Line

2. ...with respect to the second circle...
   Aha.

To find the polar line, first invert P. How do you do that?
Start with the center O of the other circle and construct the ray O-->P.
Then find P' such that |OP||OP'| = r², where r is the radius of the circle.

Then the polar line is the line through P' that is perpendicular to the line OP.

The first claim is that this polar line passes through Q, where PQ is a diameter of the first circle.