F.5 AMATHS

1, A circle C cuts the x-axis at A and B, and the y-axis at P and Q. it is given that AB=2 and major arc PQ:minor arc PQ =3:1. the distance from the center G of the circle C to the line x-2y=0 is d=1/ √5 find the equation of circle C.
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2, three pt. A(0,-14) B(-5,1) and C(7,-5) are given. A perpendicular is drawn from A to the line BC. the foot of perpendicular is D.
a) find the equation of the circle passing through the pt. A,C and D.
角 ADC = 90 deg.
AC is diameter of circle
mid-point = centre = (7/2, -19/2)
diameter = AC = sqrt(7^2 + 9^2) = sqrt(130)
circle :
(x - 7/2)^2 + (y + 19/2)^2 = [sqrt(130)/2]^2
(x - 7/2)^2 + (y + 19/2)^2 = 65/2

b) find the ratio of the areas of △ADC and △ADB.
slope of BC = 6/-12 = -1/2
slope of AD = 2
eqt. of AD : y + 14 = 2x
eqt. of BC : y - 1 = (-1/2)(x + 5)
solving, x = 5, y = -4
D = (5, -4)
area of ACD = (1/2) 乘 下面舊野
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= (1/2)(-28 - 70 + 98 + 25)
= 25/2
area of ADB = (1/2) 乘 下面舊野
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= (1/2)(5 + 70 + 70 - 20)
= 125/2
ratio of the areas of △ADC and △ADB
= 25/2 : 125/2
= 1 :５