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If the chord of contact of the tangents drawn from a point on the circle `x^2+y^2=a^2` to the circle `x^2+y^2=b^2` touches the circle `x^2+y^2=c^2` , then prove that `a ,b` and `c` are in GP.

Text Solution

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Let (h,k) be the point `x^(2)+y^(2)=a^(2)`. Then,
`h^(2)k^(2)=a^(2)` (1)
The equation of the chord of contact of tangents drawn from (h,k) to `x^(2)+y^(2)=b^(2)` is
`hx+ky=b^(2)` (2)
This touches the circle `x^(2)+y^(2)=c^(2)`. Therefore,
`|(-b^(2))/(sqrt(h^(2)+k^(2)))|`
or `|(-b^(2))/(sqrt(a^(2)))|` [Using (1)]
or `b^(2)=ac`
Therefore, a,b, and c are in GP.
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