Edexcel IAL WMA12/01/P2 /June 2022/Q10 (Equations of Circles, Tangents, Discriminant)
The circle C has centre X(3, 5) and radius r
The line l has equation y = 2x + k , where k is a constant.
Show that l and C intersect when
(3)
Given that l is a tangent to C,
show that 5r2 = (k + p) 2 , where p is a constant to be found.
(3)
The line l
cuts the y-axis at the point A
touches the circle C at the point B
as shown in Figure 2.
Given that AB = 2r
find the value of k
(6)
SOLUTION
a-
We will solve the both equations (equation of circle and line) simultaneously to show that the given expression when both circle and line intersects. 
Since the circle C has centre X(3, 5) and radius r, the equation of circle will be
Now, substituting the value of y from the equation of line into the equation of a circle (
b-
Since the line l is tangent to C therefore the above equation, we found would have only one solution meaning thereby its discriminant would be equals 0.
So the equation of their intersection found in earlier part is
Where,
c-
ABC forms a right angle triangle, applying the pythagoras theorem.
On the otherhand, if we find the distance formula to find the distance between point A
On equating both equations (1) and (2),
Whereas we derived an expressiion in part (b) as
Now, equating equation (3) and (4), we get
