When two circle touches each other orthogonally in polar coordinates?
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To find g1g1 and g2g2 you must use the tangency condition: substituting y=mx+cy=mx+c in the circle equations, the discriminant of the resulting equations must vanish. This leads to
(mc+g)2−(1+m2)(c2−k2)=0.(mc+g)2−(1+m2)(c2−k2)=0.
I wrote simply gg because the two equations are identical, so that g1g1 and g2g2are just the two solutions of the above equation. From that equation you can then immediately find g1g2=(m2+1)k2−c2g1g2=(m2+1)k2−c2 and equating that to −k2−k2 you get the required relation
(mc+g)2−(1+m2)(c2−k2)=0.(mc+g)2−(1+m2)(c2−k2)=0.
I wrote simply gg because the two equations are identical, so that g1g1 and g2g2are just the two solutions of the above equation. From that equation you can then immediately find g1g2=(m2+1)k2−c2g1g2=(m2+1)k2−c2 and equating that to −k2−k2 you get the required relation
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