Circle Geometry (1 Viewer)

[wantingtoknow]

The two questions I need help are from Year 12 Cambridge 3U, page 376 Q18.

Question is: Prove the following general cases:
a) The direct common tangent f two circles touching externally is the geometric mean of their diameters (ie the square of the tangent is the product of the diameters).
b) The difference of the squares of the direct and indirect common tangents of two non-overlapping circles is the product of the two diameters.

Also, would anyone happened to have worked solutions to this textbook? I'm constantly getting stuck on questions

Thanks.

 

[Drongoski]

I'll attempt Part (a) first.

Draw 2 touching circles, the larger one, centre P, with radius 'R' and the smaller one, centre Q, radius 'r'. (Make the 2 circles sufficiently different in radii for your diagram) Let the points of contact of one external (direct) tangent be A and B respectively. Let D be foot of perpendicular from Q to radius AP. Then the direct common tangent AB = QD since ABQD is a rectangle. Since PDQ is a right-angled triangle, QD^2 = PQ^2 - PD^2 = (R + r)^2 - (R - r)^2 {a diff of 2 squares}
= (R + r + R - r) (R + r - (R - r)) = 2R x 2r

i.e. AB^2 = 2R x 2r = product of the 2 diameters. Hence the required result.

Q.E.D.

 

[wantingtoknow]

Wow, thanks oO" How did you come up with that?

EDIT: How does QD = R + r? D isn't the centre of the circle. You don't know the distance from D to the circumference, right?

 

[Drongoski]

No No No !! QD is not equal to R + r. QD is perpendicular to AP and PQ is the hypotenuse of triangle PDQ. PQ = R + r. You may legitimately ask this: let the point of contact of the 2 circles be E. Are you sure P, E and Q are collinear ?? i.e are you sure PEQ is a straight line ??

Well, PE and QE are perpendicular to the common tangent to the 2 circles @ E . Therefore the 2 angles PE and EQ make with this tangent (being 90 degrees each) are supplementary; thus PEQ is a straight line.

(I'm sorry my response is so wordy because I don't know how to use the various new aids like LaTeX.)

 

[wantingtoknow]

Ohh, I see. Thanks so much. How did you know you have to do that? :S

 

[Drongoski]

Don't understand your question. Please elaborate.