A.maths 高手請答

(1) 4 points are concyclic means that:
We can find a circle such that all these 4 points lie on the circumference of th circle.
Therefore, we may first let the equation of the circle be:
x2 + y2 + Dx + Ey + F = 0
Then, sub any 3 of the given points into the equation, we have:
4 + 4 + 2D + 2E + F = 0
2D + 2E + F = -8 ... (1)
1 + 1 + D + E + F = 0
D + E + F = -2 ... (2)
25 + 1 + 5D + E + F = 0
5D + E + F = -26 ... (3)
Solving, we have:
D = -6, E = 0 and F = 4
Therefore the equation of the circle is:
x2 + y2 - 6x + 4 = 0
Sub the remaining point (4,-2) into the equation:
42 + 22 - 6×4 + 4 = 0
So this point also lies on the circle and hence the 4 points given are concyclic.
(2) Let the equation of the tangent be y = mx where m is in fact the slope of the tangent and it will have 2 possible values.
Using the fact that perpendicular distance from centre to the tangent is the radius of the circle:
Centre of the circle = (2,3)
Radius of the circle = 2
Rewriting the equation of the tangent as mx - y = 0
Therefore:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/May07/Circle8-1.jpg

One of the tangents has a slope of 5/12.
And the other will be the y-axis since the perpendicular from (2,3) to the y-axis is 2, the radius of the circle.
Therefore:
tan (90° - θ) = 5/12 (the angle between the line of slope 5/12 and the x-axis is 90° - θ)
θ = 67.4°

2007-05-17 01:15:21 補充：
θ = 67.4°tan θ = 12/5簡單一點:tan (90° - θ) = 5/121/tan θ = 12/5tan θ = 12/5

your first question i have answered to u


2)

let y = mx be the tangent

mx - y = 0

C: x^2 +y^2 -4x -6y+9=0

centre:(2,3) , radius:2

|(2m - 3) / [m^2 + 1]^1/2| = 2

4m^2 - 12m + 9 = 4m^2 + 4

m = 5/12

hence , the tangents are 5x - 12 = 0 and x = 0,

hence,tan@ = 12/5

1)
即係果四點都係一個圓上面

let A = (2,2), B = (1,1), C = (4,-2) and D = (5,1)

if you draw four points on graph, and can show that angleBAD + angleBCD = 180,
you can conclude that four points are concyclic.

so, now find above two angles

slope of AB = 1
slope of AD = -1/3
angleBAD = arctan(-1/3) - arctan(1)

slope of BC = -1
slope of CD = 3
angleBAD = arctan(-1) - arctan(3)

angleBAD + angleBAD = 180

proof completed

2)
centre = (2,3), radius = 2
if you plot the circle, you know y-axis is one tangent

let A be point of contact of y-axis and circle,
O be origin,
and C be centre

so, θ = 2 * angleAOC = 2 * arctan(2/3)
tanθ = 2.4

做呢d題目一定要劃圖,一劃就睇到好多野