A Tangent Line Through a Focus

Difficulty: Hard · Topics: Geometry, Conic Sections, Ellipse

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A circle has centre A(6, 0) and radius 6. Among its tangent lines, let l be the one whose slope is 1 and whose y-intercept is positive; let P be the point where l touches the circle and let R be the point where l meets the x-axis.

graph "x":[-5.4,13],"y":[-7,7.6],"xTicks":[],"yTicks":[],"label":"A circle centred at A with tangent line l, and the ellipse with foci R and A through the point of contact P","items":["type":"circle","at":[6,0],"r":6,"type":"ellipse","at":[1.757,0],"rx":6,"ry":4.243,"type":"line","from":[-5.2,-2.715],"to":[5.1,7.585],"ink":true,"type":"polygon","points":[[1.757,4.243],[-3.899,-1.414],[6,0]],"ink":true,"fill":0.16,"type":"right","at":[1.757,4.243],"from":[6,0],"to":[5.1,7.585],"size":0.45,"type":"point","at":[1.757,4.243],"label":"P","anchor":"nw","type":"point","at":[-3.899,-1.414],"label":"Q","anchor":"se","type":"point","at":[-2.485,0],"label":"R","anchor":"ne","ink":true,"type":"point","at":[6,0],"label":"A","anchor":"se","ink":true,"type":"label","at":[5.9,7.2],"text":"l"]

The ellipse with foci R and A that passes through P meets the line l at one further point Q (Q≠ P).

Find the area of triangle PQA.