SAT1000 - P843

Geometry Level pending

As shown above, the ellipse has equation: x22+y2=1\dfrac{x^2}{2}+y^2=1, line l:y=k1x32l: y=k_1 x-\dfrac{\sqrt{3}}{2} intersects with the ellipse at point A,BA,B.

Point CC is on the ellipse and line OCOC has slope k2k_2, k1k2=24k_1 k_2=\dfrac{\sqrt{2}}{4}.

MM is a point on ray OCOC, MC:AB=2:3|MC| : |AB| = 2 : 3, and the radius of circle MM is MC|MC|, OS,OTOS,OT are two tangent lines of circle MM and S,TS,T are tangent points.

Then find the maximum value of SOT\angle SOT (in radians), and find the slope of ll when SOT\angle SOT reaches the maximum.

Let θ\theta be the maximum value of SOT\angle SOT (in radians), kk is the slope of ll. Submit 1000(θ+k)\lfloor 1000(\theta+|k|)\rfloor.

Have a look at my problem set: SAT 1000 problems


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