I'm a Year 11 Geometry & Trigonometry student in Australia. I've never really had many problems with this subject, and I get a two page assignment and an hour long test every week, and an investigation every month, and am doing pretty well. But I just cannot work out the Investigation i've just been given. And its due monday. I've spent about 5 hours on it and I keep getting different answers and none of them work out.
1. Points P and C have position vectors p and c given by p = 9i + 7j and 4i + 2j.
A circle, with centre at point C cuts the x asis at the point with co-ordinates (10,0).
Find
(i) a vector equation of this circle.
This was fairly easy. |r - (4 2)|= root40. Am I correct?
Tangent lines from point P touch this circle at points A and B.
Find
(ii) the lengths, PA and PB, of the tangents.
This was easy too, root10 for both.
(iii) vector equations for the tangent lines
(iv) the position vectors, a and b, of the points of contact A and B.
My teacher says its better to do (iv) first, because then you know the direction of the two lines, hence you can work out the vector equations. But I have no idea how i'm supposed to go about it. I'm fairly sure it involves CA.PA=0, because it involves lines perpendicular to one another. And I ended up with:
x^2 + y^2 -13x -9y + 50 = 0, then taking this equation away from the equation of the circle (x^2 + y^2 - 8x - 4y + 20 = 0).
Then the two x^2 + y^2's cancel and you can find y with some algebra, then you can put y back into the first equation to find x. But I keep getting weird things for x. I got 4 and thought it sounded ok, but my teacher says that the final answer for (iv) is (6 10), and (10 4). But I just can't get these.
Then to cap it all theres another really hard question.
2. In general, find an expression in terms of p, c and r for the length of the tangents from any point P, external to a circle, with centre C and a radius r.
Any help would be much appreciated. Thanks.