Τετάρτη, 1 Ιουνίου 2016

INVERSION - Hints and Solutions

To solve these problems, invert the figure with respect to O.
1. Points P, Q, R lie in order on a straight line, and POR = 90°. Show that the circles PQO, QRO are orthogonal.
• Line PQR maps to a circle through O with PR as diameter. The two given circles map to perpendicular segmentsPO, QR (angle in semicircle). Hence the circles are orthogonal.

2. Find the inverse of a set of coaxial circles having O as one of the limiting points.

• We get concentric circles with centre O. For the orthogonal system maps to straight lines through O.

3. Points O, A, B, X lie on a circle. A line through O cuts XA, XB in U, V respectively. 
Show that the circles OAU, OVB have a common tangent at O.

• Draw the inverse figure, and show that image segments AU, VB are parallel. (Look at the angles subtended byAO, BO.)

4. BOC is a given angle. A circle through O and B has its centre on OC, and a circle through O and C has its centre on OB. The two circles meet again in X. Show that OX is a diameter of circle OBC.

• Invert to get a kite OCXB with right angles at B and C. Now show BC pperpendicular to OX.

5. (a) By thinking about the properties of inverse points, describe where the image of a circle through O lies under inversion with respect to O.
(b) Points A, B, C lie on a straight line, and O is a point not on that line. Circles OBC, OCA, OAB have centres U,V, W. Prove that the points O, U, V, W lie on a circle. [Hint: You may find it helpful to draw an accurate diagram. Also, remember the Simson line!]

• (a) Let the circle of inversion have centre O, radius 1. If OP is a diameter of the circle to be inverted, and P maps to P' then OP.OP' = 1. If C is the centre of this circle, C maps to C' with OC.OC' = 1. OC = OP/2 implies OC' = 2OP'.

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