Let the point B be the reflection of the point $$A2$$,$$3$$ with respect to the line $$8x$$−$$6y$$−$$23=0$$. Let CA and CB be circles of radii $$2$$ and $$1$$ with centers A and B respectively. Let T be a common tangent to the circles CA and CB such that both the circles are on the same side of T. If C is the point of intersection of C and the line passing through A and B, then the length of the line segment AC is $$______$$

Question

Let the point B be the reflection of the point $$A2$$,$$3$$ with respect to the line $$8x$$−$$6y$$−$$23=0$$. Let CA  and CB be circles of radii $$2$$ and $$1$$ with centers A and B respectively. Let T be a common tangent to the circles  CA and CB such that both the circles are on the same side of T. If C is the point of intersection of C and the line passing through A and B, then the length of the line segment AC is $$______$$

Infinity Learn · Accepted Answer

Centre of the circles $$CA=2$$,$$3$$ and CBPerpendicular distance from A to the line  $$8$$𝑥−$$6$$𝑦−$$23=0$$ is AD $$=$$ $$82-63-2382+62=$$ $$52$$ ∴$$AB=2$$ $$AD=5$$ and suppose that $$BC=x$$ ∴$$5+xx=21$$⇒$$x=5$$    $$($$  from the similar triangle $$property)$$∴$$AC=5+x=10$$

Detailed Solution

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