Question

If the inversion of a point A(3, 4) with respect to a circle
having centre as origin is A'(12, 16), find the diameter of the
circle. Also find the inversion point of B(6, 8) with respect to
the circle.
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Answer 1

Answer:

What is the relative inversion point of the point (4,5) of a circle with a centre of (2,3) and radius of 4 units?

What is the relative inversion point of the point (4,5) of a circle with a centre of (2,3) and radius of 4 units?

I don’t know what you mean by “relative”.

The inverse point has to be on the ray from (2,3) through the point (4,5) . And the product of the distances from (2,3) of the points (4,5) and the inverse point must be the square of the radius. So, if the point is (x,y) you must have (4−2)(y−3)=(5−3)(x−2) and also ((x−2)2+(y−3)2)((4–2)2+(5–3)2)=(42)2 . (I squared the distances to avoid taking square roots.)

So y−3=x−2 and 2(x−2)2=32 . Therefore x−2=4 i.e. x=6 and y=7 . So the inversion point is (6,7) .