一條關Circle既問題

ai) Find the coordinates of the centre A.
x²+y²+4x-6y-3=0
x²+4x+4+y²-6y+9=3+4+9
(x+2)²+(y-3)²=4²
Centre A is (-2,3)

aii)Find the radius of C1.
radius is 4

bi)Find the distance of AB.
AB = square root of ((4-(-2))²+(11-3)²) = 10

bii)Hence,find the radius of C2.
since two circles touch each other,
AB = radius of C1 + radius of C2
radius of C2 = 10-4 = 6

c)Find the equation of C2 in general form
(x-4)²+(y-11)² = 6² = 36
x²+y²-8x-22y+101 = 0

d)Find the equation of the equation of the common tangent passing though P.
Let P be (a,b )
AP/BP = 4/6 = 2/3
a=(-2x3 + 4x2)/(2+3) = 2/5
b=(3x3 + 11x2)/(2+3) = 31/5
P is (2/5, 31/5)

slope of AB = (11-3)/(4-(-2)) = 8/6 = 4/3
slope of AB x slope of tangent = -1
slope of tangent = -1/(4/3) = -3/4

by point slope form, eqn. of tangent is
(y - 31/5)/(x - 2/5) = -3/4
(5y - 31)/(5x - 2) = -3/4
4(5y - 31) = -3(5x - 2)
20y - 124 = 6 - 15x
15x + 20y = 130

ai) A = (-4/2,-(-6)/2) = (-2, 3)
aii) Radius = sqrt(4^2/4+(-6)^2/4-(-3)) = 4
bi) Distance of AB = sqrt((4+2)^2+(11-3)^2) = 10
bii) Radius of C2 = 10 - 4 = 6
c) Equation of C2:
(y-11)^2 + (x-4)^2 = 36
x^2 + y^2 - 8x - 22y + 101 = 0
d) Coordinates of P: ((2*4-3*2)/5,(2*11+3*3)/5) = (2/5,31/5)
Differentiate C2 with respect to x,
2x + 2y(dy/dx) -8 -22(dy/dx) = 0
dy/dx = (8-2x)/(2y-22) = (4-x)/(y-11)
Sub P, dy/dx = -3/4
Therefore the line is (y-31/5) = -3/4(x - 2/5)
5y-31 = -3/4(5x-2)
20y-124 = -15x+6
15x+20y-130=0
3x+4y-26=0

x+y+4x-6y-3=0
a(i)
x+y+4x-6y-3=0
=> (x+2)+(y-3)-3-4-9=0
=> (x+2)+(y-3)=16
=> (x+2)+(y-3)=42
The centre A is (-2, 3)
a(ii) radius, r = 4
(b)(i) distance AB = [(-2-4)2 + (3-11)2]0.5
= (36 + 64)0.5
= 10 units
(b)(ii) radius of C2 = AB - radius of C1 = 10 - 4 = 6 units
(c) (x - 4)2 + (y - 11)2 = 62
x2 - 8x + 16 + y2 - 22y + 121 = 36
x2 + y2 - 8x - 22y + 101 = 0
(d)
slope of AB = (11 - 3)/(4 + 2) = 4/3
slope of the tangent = -1 / (slope of AB) = -3/4
The coordinates of P : [(4x4+(-2)x6)/10, (11x4+3x6)/10]
= (0.4, 6,2)
The equation of the tangent:
(y - 6.2) / (x-0.4) = -3/4
4y - 24.8 = -3x + 1.2
4y + 3x - 26 = 0