From a point A common tangents are drawn to the circle $$x2+y2=a22$$ and the parabola $$y2$$ $$=$$ $$4$$ ax. If the area of the quadrilateral formed by the common tangents, the chords of contact of the point A, w.r.t. the circle and the parabola is λ square unit, then the value of $$256a2$$λ must be

Question

From a point A common tangents are drawn to the circle $$x2+y2=a22$$ and the parabola $$y2$$ $$=$$ $$4$$ ax. If the area of the quadrilateral formed by the common tangents, the chords of contact of the point A, w.r.t. the circle and the parabola  is λ square unit, then the value of $$256a2$$λ  must be

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Here, centre of the circle is the vertex of the parabola and both circle and parabola are symmetrical about axis of parabola . in this case the point of intersection of common tangent must lie on the directrix and axis of the parabola.i.e, $$A-a$$,$$0chord$$ of contact of circle w,r,t   $$A-a$$,$$0$$ $$isx-a+y.0=a22$$∴$$x=-a2coordinates$$ of R is $$-a2$$,$$a2$$ and  chord of contact of  parabola w.r.t $$A-a$$,$$0$$ $$isy.0=2ax-a$$  i.e $$x=acoordinates$$ of P is $$(a$$,$$2a)Area$$ of quadrilateral PQRS' $$=2Area$$ of ∆$$PAS-Area$$ of $$RAN=212.2a.2a-12$$.$$a2a2=4a2-a24=154a2=$$λ  $$256a2$$λ$$=960$$

From a point A common tangents are drawn to the circle x2+y2=a22 and the parabola y2 = 4 ax. If the area of the quadrilateral formed by the common tangents, the chords of contact of the point A, w.r.t. the circle and the parabola is λ square unit, then the value of 256a2λ must be

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