Given that the circle C has center $(a,b)$ where $a$ and $b$ are positive constants and that C touches the $x$-axis and that the line $y=x$ is a tangent to C show that $a = (1 + \sqrt{2})b$
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i think it might be Pythagoras, because the tangent means a 90 degree angle, but i can only get 2 sides and i cant get the last side so im not sure – John Smith Sep 14 '14 at 13:53
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Draw the circle showing the two tangents, then you may notice some symmetry about the line passing through the origin and the center of the circle. – SleuthEye Sep 14 '14 at 13:59
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Your circle has to be tangent both to the $x$-axis and to the line $y=x$. These two straight lines form 4 angles, and the center of the circle lies on the bisectrix of the (only) angle which is contained in the first quadrant. This is because the distance of the center from both lines is the same (the distance from the center of a circle to a line tangent to the circle is the radius). From here it should be downhill...
Without using trigonometry, you can also consider the point $A$ of tangency between $y=x$ and the circle. Take the line $r$ which passes through A and the center of the circle, and call $B$ the intersection of this line with the $x$-axis. Call $O$ the origin; what can you say about the triangle $OAB$?
marco trevi
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Easy to solve using simple geometry, write sin (45º)
Akila Uyanwatta
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Thanks ! I almost understand, but how is sin(45) = (rsin45 + r)/a ? – John Smith Sep 14 '14 at 14:39
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Both AB and AC are tangent to the circle, so AC is also a. BC = r + r.cos45 – Akila Uyanwatta Sep 21 '14 at 10:32