As shown in the figure, in the plane rectangular coordinate system, the parabola passes through a (- 1 0), b (4 4,0), c (0 0,4), ⊙M is the circumscribed circle of △ABC, and m is the center of the circle.
(1) Find the analytical formula of parabola;
(2) Find the area of the shadow part;
(3) There is a point P on the positive semi-axis of X-axis, let PQ⊥x-axis intersect BC at Q, let PQ=k and the area of △CPQ be S, find the functional relationship between S and K, and find the maximum value of S. 。
Solution: (1) passes through a parabola (-1, 0), b (4 4,0),
Let the analytical formula of parabola be: y=a(x+ 1)(x-4),
Substituting C(0, -4) into the above formula gives -4a=-4 and a = 1.
∴y=(x+ 1)(x-4)=x^2-3x-4.
(2)∫A(- 1，0)，B(4，0)，C(0，-4)。
∴OB=OC=4,OA= 1
∴∠OBC=45 ,∴∠AMC=90
∴am^2+mc^2=oa^2+oc^2= 12+42= 17
∴AM^2=CM^2= 17/2，
∴S shadow = 17/8 π.
(3) ∠ OBC = 45, PQ⊥x axis;
∴BP=PQ=k，
∴S= 1/2k？ (4-k)=- 1/2k^2+2k.
When k=2, Smax = 2.