alternate common tangent.
by hemangsarkar, Sep 6, 2012, 6:28 AM
Q) same question as before, just replace the "direct" with "alternate".
solution -
let the alternate common tangent intersect at a point
, on the 
axis between the circles.
and let the centers be
and 
.
let the radii be
an 
.
let the tangent touch the circles at
and 
.
we want to find out
using pythagoras theorem,
$KL = \sqrt{\left(\frac{r_{1}d}{r_{1} + r_{2}} \right) - r_{1}^2 } + \sqrt{\left(\frac{r_{2}d}{r_{1} + r_{2}} \right) - r_{2}^2 } = \sqrt{d^2 - (r_{1} + r_{2})^2}$
solution -
let the alternate common tangent intersect at a point
, on the 
axis between the circles.and let the centers be
and 
.let the radii be
an 
.
let the tangent touch the circles at
and 
.
we want to find out
using pythagoras theorem,
$KL = \sqrt{\left(\frac{r_{1}d}{r_{1} + r_{2}} \right) - r_{1}^2 } + \sqrt{\left(\frac{r_{2}d}{r_{1} + r_{2}} \right) - r_{2}^2 } = \sqrt{d^2 - (r_{1} + r_{2})^2}$
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