ce a.maths CO-GEOM(CIRCLE) 3

Andrew

2007-12-04 5:33 am

ce a.maths CO-GEOM(CIRCLE)

1. Find the common chord of the circles x^2+y^2-4y-3=0 and x^2+y^2-4x-2y-5=0.
2.Find the equation of the circle which passes through the points of intersection of the line x-2y=0 and the circle x^2+y^2+4x-2y+1=0 and whose centre lies on the x-axis.
3.Prove that the circles x^2+y^2+2x-8y+8=0 and x^2+^2-6x-2y+6=0 touch each other, and find their common tangent at the point of contact.

回答 (1)

六呎將軍

2007-12-04 11:09 pm

✔ 最佳答案

(1) The common chord can be found by just subtracting the equations of 2 circles:

(x² + y² - 4y - 3) - (x² + y² - 4x - 2y - 5) = 0
4x - 2y + 2 = 0
x - y + 1 = 0

(2) The family of circles from the given information is:

(x² + y² - 4x - 2y - 5) + k(x - 2y) = 0 where k is a variable

x² + y² + (k - 4)x - (k + 2)y - 5 = 0
Its centre is located at:
[(4 - k)/2, (k + 2)/2]

For it to lie on the x-axis, its y-coordinate should be zero, i.e.

k + 2 = 0
k = -2

So the equation of the circle is:
x² + y² - 6x - 5 = 0

(3) Centres are (-1. 4) and (3, 1).
Hence distance between centres is:
√[(-1 - 3)² + (4 - 1)²]
= 5

Also, radii of the circles are:
(1/2)√[2² + (-8)² - 4(8)] = 3 and (1/2)√[(-6)² + (-2)² - 4(6)] = 2

So we can see that sum of radii = distance between centres and therefore the circles touch EXTERNALLY.

參考: Myself

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