1. The equation of a family of circles F is given by
x^2 + y^2 – 8kx – 6ky + 25(k^2 – 1) = 0
where k is real.
(a)(i) Find the centre of a circle in F in terms of k.
Hence show that the centres of all circles in F lie on the line 3x – 4y = 0.
(ii) Show that all circles in F have the same radius 5.
http://i617.photobucket.com/albums/tt257/michaelcoco_/1.jpg
(b) The figure above shows some circles in F. It is given that there are two parallel lines, both of which are common tangents to all circles in F.
Write down the slope of these two common tangents.
Hence find the equations of these two common tangents.
※ 不用計算(a) part，只供閣下參考計算(b) part之用※
※ 圖上的直線是穿過O點及圓心※
一條關於Circles的簡單A.Maths.問題
2009-04-26 11:20 pm
回答 (2)
2009-04-27 12:53 am
2009-04-27 12:47 am
Slope of the 2 tangents same as the slope of the line of centres = 3/4.
Let (h,k) be a point on the common tangent, so distance from this point to line 3x - 4y = 0 equals to radius of circle, that is
abs[(3h - 4k)/sqrt(3^2 + 4^2)] = 5.
so the tangents are:
3h - 4k = 25 or 3x - 4y = 25 and
3h - 4k = - 23 or 3x - 4y = 25.
2009-04-26 16:49:56 補充：
Correction : The last line should be - 25, not -23. And the equation is 3x - 4y = - 25.
Let (h,k) be a point on the common tangent, so distance from this point to line 3x - 4y = 0 equals to radius of circle, that is
abs[(3h - 4k)/sqrt(3^2 + 4^2)] = 5.
so the tangents are:
3h - 4k = 25 or 3x - 4y = 25 and
3h - 4k = - 23 or 3x - 4y = 25.
2009-04-26 16:49:56 補充：
Correction : The last line should be - 25, not -23. And the equation is 3x - 4y = - 25.
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